The first thing that I noticed about Mike is that his style of speaking makes him seem rather unintelligent most of the time. He tends to say sort of stupid, extremely obvious things that don't need to be said, and, more prominently, he repeats things a huge amount. He calls Brett "a lovely piece" a total of five or six times in a stretch about a page long the first time we meet him, and keeps doing this kind of thing from then on. This repetition may be in part a result of him being drunk, which he pretty much always is in the book.
Even among this group of privileged party-goers, Mike stands out from the rest as not having a real purpose in life. He's always drunk, he no longer has a job, and he doesn't seem to have any real ambition or passions; he's just along for the ride with Brett. He also seem to be almost comically bad at everything he does; the first thing that we hear about him is that he got injured while attempting to help an old woman with her bags. This is a man who manages to go broke both gradually and suddenly at the same time. Brett also asks him to tell a bunch of stories about the war that are presumably about his incompetence; he doesn't want to tell them because they "reflect discredit" on him.
Brett's relationship with Michael is very interesting. Brett seems to be at least fond of Mike, and seem him as a good friend, but it's clear that she is at least in part marrying him for his family's money. She's not really emotionally attached to him, and has no discomfort with pursuing other men, even with Romero where Mike is there with her. Mike is aware of these relationships; Brett freely tells him all about them. He generally doesn't seem to have a problem with them; he understands the nature of the relationship and seems satisfied that he's going to be able to marry such "a lovely piece." Brett and Mike "understand each other." Cohn, however, is one man that's had an affair with Brett that he can't stand, because he's still hanging around her afterwards.
Mike is extremely hostile to Cohn, and even tries to fight him a couple times. He might have problems with Cohn because of his annoying personality and superior attitude, but I think that his main anger with Cohn is more because of what he represents. Before, he could sort of ignore the affairs that Brett had, but it's impossible to now that Cohn is physically there with them. He's thinking more about the nature of his relationship with Brett, and he's perhaps getting insecure, which is making him angry. It'll be interesting to see if Mike and Cohn eventually have a real fight over Brett, as I think that the story's leading there.
Mike seems to generally get along with the rest of the group, but there's not any special friendship there like there is between Jake and Bill. They seem to see him as a pretty cool guy, one that's fun normally but can get very annoying when too drunk. Jake, Bill, and Brett also all support his attacks on Cohn to some level. Mike, on the other hand, really wants to be fully part into the group and be one of the guys. From his checking several times to make sure they don't mind him coming on the trip to his pausing to make sure everyone's laughing during his stories, it's clear that he craves acceptance. This most evident when he's on the verge of tears while trying to get Jake to side with him about Cohn. Mike just isn't as "cool" a person as the rest of the group it seems; he's desperate to be liked and to it in but he's sort of left out.