It's quite difficult to put some information in an answer without hiding it, but I'll limit it to a minimum. From screenrant:
Despite Mike not going to Harvard or any other law school in Suits, he became one of Pearson, Hardman's best associates thanks to a series of events.
It's mentioned on another answer on this very same stack that:
In a third season flashback episode, it is revealed that Mike was accepted into Harvard Law ten years before the series began.
From a description of Season 3 episode 6:
Mike confesses to his role in the test selling to the dean. It turns out his daughter bought the test from Mike's friend and he's being forced to resign. As payback he's going to expel Mike and make sure he doesn't go to Harvard.
More information (and spoilers!) on Netflix
After Mike is arrested and ADA Anita Gibbs (Leslie Hope) begins building her case against him, he learns that someone from Harvard sent her an anonymous email. At first he believes it’s Henry Gerard, an ethics professor who knows Mike’s secret. Later, the firm gets their hands on the letter and Louis recognizes the language. Sheila Sazs wrote it.
Louis dated Sheila (Rachael Harris), a Harvard employee, off and on. After Mike’s promotion, she saw an article about him attending Harvard. When she couldn’t find records to prove that, she sent in the tip.