notes and warnings: Premiere will not run too well on low end systems. Look in my a-m-v.org profile for what my system looks like. I rank it quite high for editing. My previous system was a 300MHz with 64M SDRAM! AND I STILL EDITED!!! ^_^
If your system is too slow to run Premiere within the limits of your patience/time and you are too cheap to upgrade/switch to another editor (I use Dazzle's MovieStar mostly, despite its ghetto reputation; thinking of upgrading from their MPEG1/2 toolset to DV), it is possible IN THEORY (currently working on the practice part) to make an AMV solely using Avery Lee's free (as in beer and as in speech) video processing tool VirtualDub, which most people around anime online should be familiar with. You'll be limited with regard to your effects, but the only things you'll really be missing from your toolbox are layering, morph, and transition effects.
If you've got the patience, you can do all kinds of transforms in VDub by editing frame-by-frame; so far I've done a 360-degree rotation, a pendulum swing, and an inversion strobe. Want a blinding strobe? Clip a single frame, use the built-in threshold filter to make it completely white, and insert that after every frame in the clip you want to flash. Want to make your video go twice as fast? Delete every other frame (1234 -> 13). Twice as slow? Dupe all frames (1234 -> 11223344). It takes time, patience, and trial and error to make VDub work for you, but that's true of any editing system.
For more advanced effects, you might want to try MEIMI (sorry I don't have a link), especially if you have Japanese fonts installed -- there's no English patch yet. I haven't had much time to work on getting it running, but I've had a video to finish lately. If anyone's remotely interested I will provide aditional details as the Freeware AMV Project proceeds.