To make your question short, and to see if I understood it correctly, we're talking about a player who made his character a certain one and roleplays it entirely different. You added that you think that it comes from inexperience, and that he created this character after you said "no" to some "freak-character"-ideas. You want to help him roleplay the character he created.
As I see it, this problem is made from two smaller ones. The first is that he doesn't see his character as interesting because the character "is normal and normal is boring". The second is that you wanna help him understand why the way he plays the character does not fit the story-world of your game.
Helping him understand that "normal is not boring
This is the more important problem, as it stands in the basis of the entire problem. If he'll see that normal characters can be interesting his "anti-persona" will perish and he'll roleplay a normal character and not a freak one. The main trick here is to show him that normal characters are not entirely normal, i.e. "no person is like the others". In order for that to work, we need to give the character depth.
The easiest way to give depth to a character is through internal conflicts. Having goals and all is nice, but without something that blocks oneself from achieving them it is far less interesting. First thing to do is to go over his character's background and see if he implemented there an internal conflict for his character. If so, show it to him and talk with him about it. If no, sit with him and help him to come with one. The internal conflict doesn't have to be extravagant, but it needs to be there. An example one might be that he loves Vincent's sister but secretly hates Vincent himself, or another like Loves the sister but thinks that he's not good enough for there. I'll take the second one as an example for this section.
The conflict gives us a few things, a few added benefits. It gives the character 2 conflicting goals: "Get the sister and prove that I'm worthy". Now, with those two we also get a kind of an achieving-plan: "If I'll show her that I'm worthy, by getting something amazing done, she'll want me and I'll be able to get her". More than that, the character gets the knowledge that each advancement in order to achieve one goal will drive the other one to the far end.
But the first conflict is even more interesting. The character here has the knowledge that he needs the brother in order to save his lover, but he just can't stand being near the brother. He'll drive the mission onward for two reasons but he'll have doubts about his lover- if he'll marry her he'll be stuck with this brother of hers.
To make long story short, simple conflicts can show the player that even normal characters are interesting and unique. When combined with goals they force the character to take certain steps along the roads, to commit certain actions along the way, that he won't want to do but will make him doubt himself and question himself and see that his problem are far more interesting than those of every freak that he'll encounter.
Another nice way to help him see the importance of conflict is through showing him and analyzing with him certain protagonists that are normal people, from the stories and movies and series (of any form)that he likes. He'll see quite quickly that the conflicts make them interesting.
But he may say that it is not enough. For that there are a few more literary tools that might help him see why normal people are interesting. The first one is having flaws (internal or external) and the second one is using "The Ghost".
Flawed characters are characters that just like normal people aren't perfect. Those flaws can be internal (self-doubts, for example, or a mild paranoia) or they can be external (they're look frightens ordinary people, for once, or a missing hand for the other). The idea is that the character has to deal with the flaw, and one day to find the strength to overcome it. The fight for the overcoming act makes the character far more interesting. A nice example of that can be seen in The Rain Man, where he learns at the end that he can count on strangers/"dumb" persons like he's brother. Another nice example can be seen in the story of The Ugly Duckling who although looking terrible learned to acknowledge himself and to accept the way he looks, to accept his difference.
"The Ghost" is an event from the past that just like a ghost haunts the character to this day. Again, trying to cope with it is what builds a deep character. One example for this can be seen in the movie Inception, where we literally have a ghost- Cob's wife. Another example for this can be seen in the movie Casablanca, where he has to deal with his broken relationship with Ilsa. This Ghost is far more interesting as the originator of the Ghost actually comes back to his life. In Frozen we see another kind of a Ghost- the act that one feels guilty about. Elsa actually killed her sister.
All of these techniques are there for one reason- to make regular people interesting, to give depth to the characters, to make them human beings with goals and drives and psychology.
Helping him see that his character doesn't fit the world
After he understands that he doesn't have to be a freak in order to be interesting, he will be far more understandable about playing a character that fits the world. Then, try to explain to him as calmly as you can what it is in the way he played his character that doesn’t fit the world.
Explain to him that the characters are in a world where being a freak is bad, where achieving one's goals is the ideal. Each and every one for himself, as the saying goes. Give him examples from the way he played his character and analyze with him, in a one-on-one conversation where his way of acting came from. Use the background he created to illustrate to him where your problem comes from.
Then ask him what problems he has with his character, and together try to find a solution. Maybe let him be just a little bit freakish. Maybe he needs to just create a different character. This is basically between you and him. After that show the updated character to the group and get their approval.
When combining those two, you'll get a player who his far more willing to both play the character while also seeing the problems with the way he played his character before.
Combining the two solutions
When combining the two solutions you get a better player, who understands for the future also how to create regular characters that are not freaks yet far more interesting than those freaks will ever be able to be. Furthermore, you get a player who is willing to play his character as written while still making the character fit into the world. Hope any of these helped you.
Energy
Put focus into roleplaying, even if you're watching someone else do it. Focus yourself on what they are saying and doing, even if it's kinda boring, and project your body language and voice while you're acting in-character. Be much less high-intensity when simply describing your bonuses while rolling, or asking someone to pass the chips. This will create natural focus on the roleplaying aspect of the game.
Speak In-Character
Wherever possible, assume what others are doing is a prompt for in-character roleplaying. DM asks what you do? Turn to someone, and say 'Philius, methinks we should cross that bridge and by Everam and St George, charge those there gnolls with swords in hand. Once we have them subdued, we'll take some answers from them, so we must leave at least some alive! What say you, well-met friend?"
Even if people don't respond in-character, and instead shift it back ooc, roleplaying just happened. Keep doing it and others will soon follow.
Ham It Up
Your character is 'quiet' and 'reasonable' and NOPE. Your character is the hammiest of ham. He's a loud cliche. He is instantly identifiable as the tropes that make him up - and he defines the setting by his very presence. It's unfortunate - but humans love ham. They love it.. a lot. Be something simple and understandable and loud, and they will get with the program really fast.
You can do this by being a masterful actor and roleplayer with any character, even a non-hammy one, but it is easiest with ham, so ham I will advise. Your paladin isn't just a paladin who likes cheese and moonlit walks on the beach - he's SIR GALAHAD THE MIGHTY, SUBDUER OF THE PEASANTS, DEFENDER OF THE WOMENFOLK, AND HIS MOUSTACHE BRISTLES AT THE SLIGHTEST SUGGESTION OF DRAGONS.
'Big' traits tend to focus things on the roleplaying a lot faster. Simpler is easier for the audience to understand.
Find Allies
Find people who will respond to your dramatic offers. When you address people, address them first, so they respond in-character, and then immediately pull other people in. People ignore offers initially, but if something is already rolling, they'll get rolled in with it. Some people will instinctively resist roleplaying offers, for all kinds of reasons - learn to identify them too, and offer to them last, once the roleplaying scene has the most momentum.
Be good at plot
Being able to identify where the adventure is going will let you advance the plot during a roleplaying scene - which both speeds up the adventure and means the time spent on roleplaying won't cause a weak GM to not let you hit the end of it.
Roleplay during combat
'LOOK OUT, FARAMIR! THE GNOLL IS AT YOUR BACK!' 'Galahad charges at the gnoll attacking' moving mini 'faramir, and' rolls dice 'swings at it with his mighty sword.' By including both speech and roleplaying-description in amongst your mechanical actions, you partially negate the disconnect that happens during the mechanics-rich combat portions of sessions. Have to know what you are doing on your turn before your turn rolls around, or anti-roleplayers will complain your roleplaying is slowing things up if you are not clearly doing it faster than anyone else.
Additionally, being good at combat, and giving tactical advice in-character that leads to defeating enemies quickly, will give more time overall for non-combat-constrained roleplaying itself.
Occasionally, roleplay during others' turns - have Galahad shout an encouraging phrase at an opportune moment. This has to be rare, and well-timed, though - an advanced technique.
Be Heroic, or Dastardly
Again, ham. By being heroic, and roleplaying it hard, you make other people who are not roleplaying feel heroic. By being dastardly, and roleplaying it hard, you make other people feel heroic also who are not roleplaying. You're giving them some of your roleplaying energy in a way that feels good for them. Morally grey is, again, a tougher sell. Note this isn't 'good' or 'evil', it's more saturday morning cartoon than that. Snidely Whiplashi, or Dudley DoRight.
Incorporate the GM
Don't just roleplay at fellow players. Roleplay at NPCs. Treat them with importance, and give the GM offers to roleplay right back at you. All of this applies to the GM, too. Getting the GM on-board with roleplaying, especially if you can advance the story while doing so, will be a tremendous boon to your cause.
You're right, by the way. Premade adventures, split up groups, schedules, public venues, this stuff just kills roleplaying and really makes it quite hard - I literally could not design a better system to do so.
But even in those kind of circumstances, I have personally sparked roleplaying in some extremely tough crowds. You won't see a huge improvement - but even the tiniest bit of roleplaying can be a huge welcome to you if you're in a roleplaying drought, and if you play with regularly the same pool of people, you'll find people gravitating to you that appreciate roleplaying, perhaps even to the extent that people will fight to have you in their groups.
Overall, though, the roleplaying will be in many ways a simpler thing than the rare high level roleplaying you can get in a home group.
But it's certainly not impossible.
Just have the courage to keep trying and don't give up.
Best Answer
One good way to get players thinking about their backstories is to pose each of them one question. For example, in my current campaign, all of my players had to tell me the answer to the question, "What one thing does your character wish for above all else?" Other good questions are, "What one thing does your character regret more than anything?", "Who would your character sacrifice his own life to save?", "How far would your character go to avoid death?", etc.
The thing all these questions have in common is that they make the character make a choice. Characters are defined by their choices more than any event or set of events in their backstories, so in order to encourage roleplaying, you need to force the characters to make difficult choices.
Note that the players don't necessarily have to tell each other these answers. In fact, you almost always see far more interesting RP when the players don't all know each others' stories already. Once the stories are out there, players often forget about them, stop playing to them, etc. They've been revealed; they're no longer interesting or relevant.
Instead, find ways to draw out characters' backgrounds and players' roleplaying bit by bit, over the course of multiple sessions. For example, instead of just dropping everyone's backstories into one session with a single trap or set of traps, you could have a door guardian who asks each character one of the choice-based questions above. The character then must decide how much to reveal - after all, if his answer to "how far would you go to avoid death?" is "become a lich if I have to", the character may not want to say that out loud in front of the party's paladin. He'll have to find a way to answer the question honestly enough for the guardian to allow him to pass, while still not giving away anything he doesn't want known - and I have years and years of experience that show that when characters know just a little information about a fellow party member, they will want more. And once that happens, you as DM can just sit back and watch the roleplay.