The Lethal Weapon movies were pretty standard buddy-cop fare, starting out gritty and trending more toward the humorous as the chemistry between Mel Gibson and Danny Glover deepened. It was the perfect role for Gibson, allowing him free reign to indulge the loose-cannon persona he cultivated in Mad Max, and would use again onscreen as Fletcher Christian and Hamlet. At least, I think it's a persona. Maybe it's really him.
The idea behind Riggs is that he's not really that interested in staying alive after the death of his wife, so he has become a lethal weapon, without regard for his own safety, drawing on skills learned during his special forces career in Vietnam to terrorize LA's criminal element. The introduction of Glover's too-old-for-this-shit Murtaugh as Riggs' partner produces the age/race/temperament dynamic that fuels both the drama and the comedy of all four films. Later, Joe Pesci and Renee Russo bring lighter, funnier support to the Lethal Weapon universe as it moves out of the hyper-realism mid-eighties into the glossier, more family-friendly early 90s.
Riggs softens, of course, throughout the series, and by the end he has become almost sane. He was more fun in the beginning.