This always seems to be the answer, but...
Talk to the player first.
I'm assuming you have some means of contacting your players outside your normal game time, if only to set up game or let each other know of cancellations or emergencies. Send your player a message, something along the lines of "Hey, I've noticed that you seem dissatisfied at game lately. Is there something going on you're not happy with? I want to make sure our game is fun for all of us, so please let me know if you have any concerns."
See how he responds to that. If he responds with a list of concerns, use those as a starting point. If they're things you can address, try to do so. If you think his problems are irreconcilable with your game and/or the other players, then say that, politely, and suggest that he might want to look for a different game to join.
If his response is neutral or brushes you off (such as "nothing's wrong, see you next game"), then you can let it go for a session or two. See how he behaves, and whether he raises the issue himself.
Ask another player to initiate communication
If you're not comfortable approaching the problem player directly (or don't have the means to do so), you can speak again with the person who told you this player wants to leave. It sounds like the go-between is friendly with him, so you can ask the go-between to, next time the player brings up the issue of wanting to leave the campaign, tell him to talk to you about it.
Don't use the go-between to actually convey messages like "I heard you want to leave, what did I do wrong?". That kind of thing usually gets lost in communication, and it puts the second player in an awkward spot. Just ask him to pass on that you're open to hearing your players' concerns and that the player is welcome to talk to you if he has a problem.
Don't jump the gun
Either way, don't rely on hearsay to drop a character and his player from the party. Don't allow the other players to sacrifice this guy's character unless you have his explicit buy-in (or if it comes up naturally in-game and the player himself is all for it). Don't assume that he meant what he said about wanting to drop, either - I've had players complain about my game in moments of frustration, but when I ask them if there's something I can do better, they reassure me they're having fun and were just briefly frustrated.
TL;DR: Communication is your friend!
Talk to the player. Follow his lead, and don't make assumptions based on what other people are telling you. If you handle this issue with grace and good will, then it's highly unlikely the other players will see a reason to stop playing a game they enjoy just because their friend did.
You're not having fun.
Since the entire point of role-playing is to have fun, you need to do something about it. In particular, it sounds like the character he's playing (called a GMPC) is seriously distracting him from his GM task of running a fun story for the other players at the table.
You have several options, depending on whether you're willing to accept him as a GM as long as he changes, or if you can accept him as a part of the group as long as he isn't GMing:
- Talk to the GM about his behavior. Try to be non-confrontational, and don't have this conversation in front of the group; that will just make him defensive and won't accomplish anything. Try persuading him to not bring a character along with the group; he's GMing, not playing.
- Sit out the current game, until the GM is done. There's no need for you to continue playing a game you dislike, so take a break until he's done running the game. Then join the next game, that someone else in the group will run.
- Take over GMing duties, with the support of the rest of the group. Since they already asked you to be the GM, they'll almost certainly be willing to let you run the game. Talk to the other players about this individually before you try this; if you have enough support, you'll be able to usurp his position as GM, and start an entirely different game. Perhaps the current GM will be a better player than they are a GM. And look on the bright side: you now have some fantastic examples of how not to GM, which will make you much better at this new role! You can also combine this with #2 above: sit out the game until it's over, then be the next GM. (Warning: given what you've described of his personality, the current GM is unlikely to take this well.)
- Start a separate gaming group, inviting whoever is willing to play without the problem GM. If some people are unwilling to play with the new group because they don't want to exclude the current GM, then you'll need to exclude them as well. Recruit a few other players from elsewhere if you need to.
- Find a new gaming group. If all else fails, just don't play with this group at all. If they're willing to play with someone they dislike, and you can't persuade them to kick out a problem GM, that's their problem. It doesn't have to be your problem.
Note that none of these options include playing a game you dislike with a GM who you hate. He isn't going to improve on his own, unless you talk him into changing or encourage the group to leave him behind.
On a slight tangent, see the Geek Social Fallacies. It sounds like some of the members of the gaming group are suffering from GSF #1: "Don't exclude anyone" and GSF #5: "Failing to invite someone is a deliberate snub."
Best Answer
Check out Belkar, from the Order of the Stick. A chaotic evil character in a mostly-good-aligned party. See this quote:
Note, though, that the OOTS keeping Belkar around has always been a bit of a stretch for willing suspension of disbelief, allowed mostly for narrative purposes, and often lampshaded by the characters themselves, so it might be a bit of a problem getting the rest of your party behind the idea of sheltering a psycho killer, but if you're fun and over-the-top enough, they might appreciate your dramatic value.
Another option that some might not like is getting explicit buy-in from the players, or at least some you think will be into it. Some might see it as a form of meta-gaming that takes away from the real roleplaying, but others might see it as a social contract that allows the party to play the roles they like and build the narratives they like.