Anti-Hero: He's an Unscrupulous Hero in the first film. The only reason he does anything is because he loves being a cop. He changes into a Pragmatic Hero in the others - he cares about his surrogate family, but he's still willing to resort to dirty tactics.
Ax-Crazy: Primarily in the first movie, mellows down in the sequels.
Bad Ass: Deconstructed in the first film. He's an extremely good soldier, but doesn't feel like he's good at anything else.
The Berserker: Even after getting over his suicidal tendencies, he still has a cavalier regard for his own survival.
Improbable Aiming Skills: Justified. He was a Special Forces Sniper in Laos and Vietnam. He once made a hit that he thinks only ten other people in the world might have been able to make.
Riggs: It's the only thing I was ever good at.
Jerk with a Heart of Gold: As much as he loves tormenting his friends, he does care about them. Best shown near the end of the fourth movie, where he has a heartfelt conversation with Leo, who he spends most of the series abusing.
Large Ham: Especially in the first movie. He's become more composed in the sequels.
"Do you really wanna jump?! DO YOU WANNA?!?"
Lightning Bruiser: In the first film, he takes out a room full of guys before they can react.
Deuteragonist: Murtaugh has virtually the same amount of screentime as Riggs, but he doesn't get the same level of character development. This is done well, since it provides a good contrast to Riggs' journey from a suicidal maniac to a committed family man.
Younger Than They Look: Danny Glover was only 40 years old when he filmed the first movie despite supposedly being "too old for this shit". Murtaugh was written as an older character (50 in the first movie).
Evil Counterpart: Like Riggs, he was an incredibly tough cop who bent the rules and got results, and they have many of the same snarky personality quirks. He's essentially a dark "what-if" version of who and what Riggs might've become if he'd continued down the self-destructive path he was on after Vicky died. Naturally, he and Riggs hate each other immediately.
Badass: Probably the most Badass character in the entire series. He takes on Riggs and Murtaugh at the same time, twice, and both times he kicks their asses. The first time, effortlessly; the second, he wants them to suffer, but even though he dies he still nearly kills both of them with his bare hands (literally) and after the fight both characters are in crutches. He kicks the ass of many other characters in the film as well. He also takes Rigg's gun apart in one swift move with one hand, and at one point he actually dodges a bullet.
Big Brother Instinct: Inverted; he spends the whole film working to buy the release of his older brother, then tries to avenge him once Riggs and Murtaugh kill him.