With today's technology, you'll be able to get a very nice computer for a reasonable price. I'll give some ideas to look for. (The video card is not as important in the editing of AVCHD video, but rather during the process of viewing the AVCHD video, for example, in fullscreen.

Some specs for a well-equipped AVCHD computer:
High-end dual core (Intel E8x00 series) or quad-core (Q6x00 series) processor
2GB RAM minimum
ATI HD3000 series or greater / nVidia GeForce 8000 series or greater
500GB minimum (7200RPM or greater) devoted ONLY for video raw files/projects/renderings