Anime
Eusine comes across as pretty arrogant when he shows up in "For Ho-Oh the Bells Toll!"—he's got this almost religious obsession with Suicune and gets visibly upset when Ash mentions he's already seen it. The jealousy is pretty obvious since Ash has also spotted Ho-Oh, something Eusine hasn't managed yet. When he finally does encounter Suicune in that same episode, his desperation to catch it reveals he's not as noble as he'd like to think. He tries to capture it mostly for his own ego, which is kind of ironic given how judgmental he is about others being unworthy of the legendary beasts. Interestingly, he's way more chill about Suicune in "The Legend of Thunder!"—he barely mentions it at all.
In "The Legend of Thunder!" he joins forces with Jimmy, Marina, and Vincent against Attila and Hun. Marina apparently develops feelings for him, which fans often chalk up to the cape.
He's apparently pretty tight with Morty, the other Ecruteak legend expert in town. Morty basically acts as his conscience, constantly trying to get him to chill out, but it never really sticks. When Eusine bails suddenly at the end of an episode, Morty doesn't even blink—suggesting this is a regular thing.
Manga
In Pokémon Adventures, Eusine's a total showboat, especially around women. He busts out flashy magic tricks and hands out signed photos of himself. He gets around on a Jumpluff and is clearly close with Morty, who helps him track Suicune.
Morty first mentions that he had Eusine use his powers to search for Suicune, but came up empty. After Morty gets beaten by Suicune himself, he reaches out to Eusine, who immediately sets off on his own hunt. That's when Eusine runs into Crystal. Things get awkward fast—Eusine's Electrode shocks Crystal's Arckee, Crystal calls him a rival, and takes off.
Later, Eusine cons an elderly couple into giving him Moomoo Milk by pretending to cure their Miltank (he actually just stuffed a berry in its mouth). Crystal catches him red-handed manipulating them, but Eusine doesn't really care. When they both encounter what they think is Suicune, it turns out to be a Ditto sent by Team Rocket as a distraction. Eusine and Crystal manage to let the real Suicune escape, and Eusine takes down the Team Rocket Elite Trio solo.
Content compiled by AnimeList.moe from publicly available sources.