Many psychical DPS champs have an alternative AP build that usually proves quite successful due to high AP scaling. But on some champions, like Tryndamere, it's not even an option. I build him starting off with a Warmogs (to max it as early as possible), then a Frozen Mallet for the HP boost and the slow, and then I stack Atma's for the added DPS and survivability. I can usually end up being an insane killing machine and an unkillable juggernaut at the same time.
As for attack speed, I have an attack speed per level rune page that gets me to around 44% I believe, and Berserker's Greaves give me the boost I need early game.