I have found that trying to maintain an exact balance was just impossible. As you said, it ends up being ridiculous: if I help an old person to cross the road, do I have to steal her bag right after that?
So after a long time trying to play this kind of character, I realized that the important part was not to maintain an hypothetical balance (which might not even exist in the first place), but to avoid extremes.
You don't want the evil demons to eradicate the Good Guys. But you don't want either the Good Guys to eradicate the evil demons.
Total war is bad. World peace too.
Civilization leading to Nature's extermination is bad. But Nature overwhelmingly destroying civilization is bad too.
It ends up being more fun to play, as you don't have to worry about any single action, only the grand scheme of things.
And I believe it is closer to the intended concept, as by making sure that both sides of the coin always exist, you are really working towards Balance.
(it will also makes you develop a "Neutral Mind", which is always a good thing)
The motives and actions you've described are not presciptively tied to any alignment; alignment is fuzzy enough that he can be justified as having any alignment you wish to choose for him.
Both fame and fortune are unaligned motivations, that is, not really solidly towards any particular alignment, including True Neutral or any other kind of Neutral. Fame and fortune factor into the goals of an enormous number of people and characters from all alignments, after all.
If you’re willing to break laws to get fame and fortune, that’s starting to look more Chaotic; if you’re willing to hurt others, particularly those who are mere bystanders, in order to get them, that’s starting to look more Evil. If your methods for achieving fame and fortune uphold the law, that’s looking more Lawful; if your methods help others, that’s looking more Good.
Someone who only upholds the laws and helps people because he believes those are the best routes to fame and fortune is clearly not a lawful good ideologue. But his actions are still squarely lawful and good. His behavior is still lawful and good. And his reasons for those actions, that behavior, aren’t really chaotic or evil.
This isn’t someone pretending to the alignment for the sake of betraying it, undoing it, or destroying it. He is not a saboteur or spy. He will, most likely, remain acting lawfully and good until the day he dies.
So what does that make his alignment? I doubt anyone could tell you definitively. The books are frustratingly vague, and worse at times self-contradictory, about what makes an alignment. Does intent matter? Many books imply that it does not, evil actions are always evil no matter what you do them, and drag you towards the “deep end of the alignment pool,” as Belkar once put it. Other books seem to imply that intention matters a great deal.
In short, no two people agree 100% about alignment, not even the authors of the books. I could take your description and make a case for any of the alignments. Here, I’ll do it:
He’s acting lawfully and to improve others’ lives. Why he does so is irrelevant, particularly when his reasons aren’t really chaotic or evil.
His actions improve others’ lives; that’s what’s important. The fact that he doesn’t have any personal belief in the law, though, means you cannot truly call him lawful.
He’s basically using the law for his own ends, making it almost a mockery of itself. But he’s still helping people.
He’s playing the game, following all the rules to a T to get what he wants. The laws, where he is, reward good behavior, so he helps people, but that’s only a coincidence of what the law happens to say.
He doesn’t really care about law or goodness; he’s just out for himself. “Looking out for number one,” isn’t evil or chaotic, though, pretty solidly neutral behavior.
Using the law while not believing a word of it is, again, a mockery. Fame and fortune are pretty neutral goals, but his cavalier attitude towards the law is chaotic.
Following the letter of the law perfectly to get what you want? That’s literally the description of Baatorian society. Sure, he doesn’t do anything evil outright, but he would and will the moment there’s a loophole that will see him receive wealth or fame for doing so.
Devils, despite their letter-of-the-law trickery, still obsess over the letter of the law, believe in it (or, more exactly, are belief in it). Not this guy; this guy is out to get his without any care for the law except insofar as it helps him, and again, only acts good because it’s in his best interest to do so. As soon as that’s no longer true....
As above, but now with more emphasis on just how much he is abusing the law.
Alignment is dumb. This is far from the only case that can be trivially argued from any of them. Sure, maybe deep down the guy has some scruples, and so the evil alignments don’t fit. And the chaotic alignments are something of a stretch, since (ab)using the law to one’s benefit is typically considered lawful behavior. You’d really have to emphasize the mockery of the law to make that stick. But the arguments are still there.
And that’s why I’ve said it before and I’ll say it again: the alignment system was invented for a game where the heroes were Good because they’re heroes, the orcs and goblins were Evil because they’re the villains, and everyone else is Neutral because they’re not involved. The dwarf was Lawful cuz he’s a dwarf, the elf is Chaotic cuz she’s an elf. That is how alignment works, that is what alignment was designed for. You can stretch it, somewhat, to go beyond that, but the farther you get from that simplistic paradigm, the more and more nine little boxes aren’t going to be enough to categorize everything and you get self-contradictory descriptions of the alignments.
In other words, the character you describe is too complex for the alignment system. Which isn’t to say he’s particularly complex – he’s not, really – but that the alignment system is extremely simplistic.
The alignment system is not very good
It doesn’t make a lot of sense, there are numerous cases where the suggestions for what is in each alignment are contradictory, it relies on the poorly-explained idea that there are objective, cosmic Goods and Evils and Chaoses and Laws. It works well enough for simple adventure-fantasy where we are the Good guys, and those are the Evil guys, and everybody not really involved are Neutral, and the Dwarf is Lawful because he’s a Dwarf and the Elf is Chaotic cuz she’s an Elf. There’s no moral question that Detect Evil cannot solve.
If you want to play a more serious game where there are shades of gray, moral and ethical questions are ambiguous and difficult to answer, and sometimes people subvert various expectations of them, then the alignment system, as written, becomes largely useless. It might make for a convenient short-hand, but with all the hang-ups and arguments that it causes, I don’t consider it useful even for that.
And the reason that all of this is the case is because the different alignments are very poorly defined. So much so that most actions and characters can make solid arguments for almost any alignment, as I do in this answer. Sometimes it seems like it’s just the action that is Good or Evil, regardless of why you do it, other times it seems like your intentions matter, and so on. Law and Chaos are that much worse, as indicated in my answer to What is a good way to explain the difference between a Chaotic Neutral character, and a character who is just crazed?. But even Good and Evil are often described in a lot of ways that make people feel they are less “good” and “evil” and more “on the side arbitrarily labeled Good” and “on the side arbitrarily labeled Evil.”
So I personally would not pay any attention to the alignment system’s perspective on this at all. If you insist, I suspect this is considered solidly Neutral rather than Evil by... most of the published books, which is to say there are still quite a few that contradict that. Unless he’s doing things that obviously just hurt people and don’t even benefit himself (stealing some dude’s medicine that he needs to survive, but which does little or no good to anyone else), it’s a Chaotic act but not an Evil act by the stupid rules.
But I’d just ignore it and figure out your own story
Behavior like this has traditionally been taken very dimly. He could (and probably would, were this some actual reality) be arrested, and in the time periods/fictional narratives that Dungeons & Dragons typically mirrors, he could expect to lose a hand, be forced into literally back-breaking manual labor for more years than he’s likely to survive, or just get hung. His party would be expected to either aid in his capture or be treated as in league with him and treated to the same punishments.
Even if he doesn’t get caught, almost all of his party should not appreciate his antics, in-character. He’s behaving in ways that are decidedly Chaotic and non-Good, which should annoy the Lawful and Good sides of the spectrum, and for everyone else he’s just likely to get them all in trouble, get them involved in some stupid investigation or arrest or whatever when they have important things to be doing (the party does have important things to be doing, right?).
In other words, there is no real good reason why his party wouldn’t kick this guy out. He only causes problems.
And you should explain all of this to the players, out of character, so they can discuss the issue. Maybe they want a silly, light-hearted game where these kinds of antics go unnoticed and they can all laugh at the ridiculous shenanigans he got up to. Maybe they don’t realize that he’s eliminating any possibility of anything more serious. Maybe they’ve all been thinking the same thing (except for the player in question), and been waiting for you to call him on it. Regardless, get input from every player on how serious a game they want, how they feel the player’s behavior is or is not derailing the game, and so on. Find a compromise or consensus, and go with that. And if everyone else disagrees with you, you have to decide whether you are going to just accept that, or find a new group. You can’t tell them they’re all wrong and make them play as you want to play.