While was never officially released for the PlayStation Portable (PSP), you can play it on the handheld using a Nintendo 64 emulator or by porting specific community-made builds. How to Play Mario Kart 64 on PSP
If you are looking for Mario Kart-style action on PSP without the hassle of emulation, the PSP has a great library of kart racers: Mario Kart 64 Psp