I'm guessing depression and alcoholism. As far as being psycho? I think that is up for discussion.
No. I think the movie pretty well expressed his dysfunction as a death wish due to the guilt over the death of his wife. I think he wasn't capable of suicide--there was a scene about this--but he wanted someone else to do the deed for him. So he kept putting himself into situations where the risk of death was great because he figured that while he still wanted to do his job, he would have been grateful had one of his foes done the job he himself could not. By the end, though, he found friendship with Murtaugh and an intense rivalry with Mr. Joshua which convinced him that maybe he wouldn't mind sticking around a bit longer.
God, Mel sure had a crazy amount of charisma back then didn't he? Just reminiscing about the movie makes me miss the time when he was one of the most exciting actors in the business. (And Busey in his prime was just as electric. I sure wish he hadn't scraped half of his brains out riding without a helmet.)
I'll buy that. If Riggs wasn't crazy then he sure had the department shrink fooled. And his methods were unconventional to say the least. Handcuffing the potential jumper then leaping onto the airbag for one. I did think that the scene where he loads the hollowpoint in his Beretta then disengages the safety was intense. I felt that though that the way Riggs took risks could have been a sign of his depression caused by the loss of his wife. I am going out on a limb and say that Riggs did have an undiagnosed illness and one that was unspecified like the one Travis Bickle did in Taxi Driver. Having a death wish is often associated with an acute mental illness. (Btw if Roger had not prevented his revolver from firing by blocking the hammer with his thumb then who knows what would have happened?)