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:
Lawful Good
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.
Neutral Good
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.
Chaotic Good
He’s basically using the law for his own ends, making it almost a mockery of itself. But he’s still helping people.
Lawful Neutral
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.
True Neutral
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.
Chaotic Neutral
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.
Lawful Evil
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.
Neutral Evil
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....
Chaotic Evil
As above, but now with more emphasis on just how much he is abusing the law.
Conclusion
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 short version of this answer is that when an eidolon crosses a life link threshold while returning to his summoner, his maximum hit points increase but his current hit points do not change, even though he lost current hit points when he crossed that threshold while moving away from his summoner. You can even kill an eidolon by bull rushing, pulling, pushing, or repositioning it (or moving it magically) across a threshold repeatedly. To directly answer your question in the comments about healing an eidolon, healing is never more effective while the eidolon is far from its summoner, and could be less effective (if the healing would bring the eidolon over its reduced maximum hit points).
Since damage should be kept track of instead of counted down thats fine.
This is an incorrect assumption, and is what's causing the confusion regarding life link. Nonlethal damage is tracked separately, but normal damage is subtracted from your current hit points. It's ok if you decide to track it as "damage taken" (and quite a bit easier, I often do so for creatures besides eidolons), but sometimes an ability needs to use the actual current hit points for something, like life link does, and it's important not to get confused: "damage taken" isn't a thing the game normally tracks, only current and maximum hit points; which means your "damage taken" marked might have to be altered to fit the new game state. Keeping this in mind, I'll explain an example, telling you how much damage the eidolon should have marked on his sheet if you want to keep tracking his hp that way, but I assure you it's much easier to simply track the eidolon's current hp (subtracting from it when he takes damage and adding to it when he is healed) than to track damage taken if he's going to be crossing life link thresholds.
Starting from an eidolon with 100 maximum hp and 44 current hp (that is, having 56 damage marked on his sheet) and within 100 feet of his summoner:
When the eidolon moves beyond 100 feet from his summoner, his current hp will drop by half to 22, and his maximum hp will drop by half to 50. You should either scribble a 50 near his max hp on his sheet and erase all but 28 of the marked damage (to show he's at 22 of 50 hp), or you should put a circle around 50 of the marked damage (to represent damage that can't be healed, since it's actually max hp loss) and increase his total marked damage to 78, to represent him being at 22 out of 100 hp.
When the eidolon then moves beyond 1,000 feet of his summoner, his current hp will drop by half again (a total loss from normal of 75%) to 11, and his maximum hp will drop by half again to 25. Again, you should either scribble a 25 where the 50 was near his max hp on his sheet and erase all but 14 of his marked damage (to put him at 11 of 25), or you should draw a second circle around another 25 damage marks (making 75 damage marked off as max hp loss) and increase his total marked damage to 89 (so it's clear he's at 11 hp from his normal max of 100).
When the eidolon then returns to within 1,000 feet of his summoner, his current hp will not change and his maximum hp will increase from 25 to 50. You should either erase the 25 and put back the 50 and increase his marked damage to 39 (to show he's at 11 of 50), or else erase the circle around the 25 damage marks and leave his total marked damage at 89.
To answer your side question about healing the eidolon while he's far away from you, the most you could heal him to at this range would be 25 hp. You'd either end up with a scribbled-in 25 max and no damage marked, or else 75 damage marked and circled and no other marks. When he came within 1,000 feet like this, you'd either change his scribbled-in max to 50 and mark 25 damage (to show he's at 25 of 50), or else just erase the circle around the 25 damage and leave him with his total marked damage at 75.
When the eidolon then returns to within 100 feet of his summoner, his current hp will not change and his maximum hp will increase from 50 to his normal max of 100. You should either erase the scribbled-in 50 and increase his marked damage to 89 (so he'll be at 11 of 100), or erase the circle around the the 50 damage marks and leave his total marked damage at 89.
The "healed" eidolon returning to within 100 feet of his summoner would either have his scribbled-in 50 erased and his marked damage increased to 75 (so he'll be at 25 of 100), or else just the circle around the 50 damage erased and his total marked damage left at 75.
Two quick examples of how nonlethal damage interacts with life link, since it is tracked separately (not as a part of current hp). 1: Take the above example eidolon with 44 hp and 100 max hp, but also with 15 points of nonlethal damage. When the eidolon crosses the first threshold, his current hp drops to 22 (and his max to 50), but he still has 15 points of nonlethal damage, meaning he's now much closer to being knocked unconscious than he was. If the eidolon then crosses the second threshold, he'll drop to only 11 hp (and 25 max hp) and 15 nonlethal damage, meaning since he has more nonlethal damage than current hit points, he immediately falls unconscious.
2: Imagine an eidolon with 100 current and maximum hit points (that is, he is unhurt, except...) with 51 nonlethal damage done to him. When he crossed the first threshold, he would drop to 50 current and maximum hit points. Since his nonlethal damage is higher than his maximum hit points, the excess point would convert to normal damage, leaving him with 49 current hit points, 50 maximum hit points, and 50 points of nonlethal damage. He would also fall unconscious. If someone carried him past the second threshold (or if his summoner walked away, beyond 1,000 feet, whichever), he would drop to 24 current and 25 max hit points. The 25 points of nonlethal damage beyond his new maximum would convert to normal damage, bringing him to -1 hit point and banishing him back to his home plane. (It's an individual GM's call as to whether this damage conversion counts as damage the summoner could sacrifice hit points to in order to prevent the eidolon from being banished.)
Best Answer
You can't do that.
Unless you mean the Unchained Summoner, which is a different class, and the eidolon can be a step away.
In that case, the answer is largely contained in In Pathfinder, does the summoner or the GM control the eidolon? and otherwise is like having any other follower, companion, etc. that is or is not the exact same alignment as you.