When Percy sends his Christmas presents back unopened in ootp, Fred and George leave their sobbing mother to Remus to look after and run off to Ron and Harry to wait until he's managed to cheer her up.
Just living with Sirius in Grimmauld Place is a massive act of kindness. He has his own home (we know - because Sirius lay low there), he has access to instant travel - there is no need for him to live at headquarters, but he chooses to move into a gloomy, cobwebby, dark magic filled mausoleum of a house with a house elf and a portrait that calls him a "filthy halfbreed", just to keep Sirius company.
Not only does he not show any resentment towards Fenrir Greyback, he also apparently bears no ill will to his father for the role he played in Remus's getting bit.
And he doesn't resent Snape for outing him as a werewolf to the entire school - saying Snape could have done a lot more damage by tampering with the potion. Remus is thankful for what Snape did right and doesn't dwell on how Snape harmed him.
in POA he comes across Harry all alone during the Hogsmeade weekend and invites him in for a cup of tea because he can tell he is lonely and frustrated. And he defies Harry's expectation of him (and all adults) by telling Harry the truth about why he didn't let him fight the boggart.
He gives up his evenings and weekends to teach Harry advanced magic, not because he knows Harry will have to fight off a swarm of dementors later in the book, not because he thinks it will save Harry's life - but so that Harry can feel more confident going into his next Quidditch game without worrying about falling off his broom.
While providing this extra curricular activity for free, in his own spare time, he also gives Harry chocolate and even a butter beer - Remus is poor; his clothes are shabby, his possessions old and battered; his teacher's salary might be good, but he knows the job isn't forever - he has one year, 12 pay packets, and then he doesn't know how long he will have to go before he finds another job. He may have to make the money he earns here last him a long time (and his even more shabby appearance in ootp, in conjunction with Umbridge's antiwerewolf laws, suggest he did) and yet he is still spending his money on things to help Harry with the side effects of dementor fighting and rewards for him.
Canon Remus is a genuinely lovely guy who looks out for others and tries to be fair and, if asked, will be generous with his time and even his money (what little he has). He doesn't put himself forward or make the first move (he would see that as him "imposing" himself on people) but when he is around, he is good to everyone and people know they can rely on him.
When Harry hears that (fake) Moody took Neville for a cup of tea, praised him over his Herbology talent and gave him a book he thought would interest him, Harry thinks "it was the sort of thing Professor Lupin would have done". Harry's response to hearing about someone doing something tactful and kind is "yeah, Lupin would have done that" - and coming from the hero of the series, who is himself kind and compassionate and thoughtful - that's pretty high praise indeed.