CPU, RAM and Hard Drive speed are the key issues.
As for what will matter to you all depends on the type of AMVs you intend on making.
CPU
If you plan on making an AMV with some nice rendered effects, Definately go for a fast CPU, an Intel Pentium 4 or equivalent if possible (look at the AMD die hard recommending Intel lol).
Pentiums are generally faster at media orientated tasks such as encoding audio and video, and if you plan on doing it regularly or extensively, then it's important to make the right choice.
RAM
As for RAM, there are a few ways to look at this. Firstly consider the software and Operating System you will be using, that will be the first point of concern, are you going to be using a relatively light OS and Non linear editor, or are you using one of the memory hungry fancy apps (something like Premiere Pro or After Effects). Then think about the complexity of the AMV you want to produce, do you want to have 1 or 2 video layers, or are you going to have multiple video layers with Image masks effects, fades and so on. Also keep in mind the size of the source files. Larger files (such as Huffyuv video) will occupy more RAM than say an MPEG2 video (which if you are using AVIsynth, depends on the filters and operations involved).
Harddrive
This is also a key component and depending on the size and speed of the drive, it will affect your editing ability considerably.
First off with harddrive size. As with RAM it depends on how many source videos you want to use, the duration, bitrate etc. You can do as many people here do and simply rip the VOBs from the DVD, and frameserve them with AVIsynth, the drawback to that is what you gain in harddrive space, you lose in CPU power (since you will be processing, IVTC'ing etc. videos in realtime in order to save space).
DVDs are usually 4 - 8 GB in size (a rough estimate!) The DVDs seem to be split into 1GB chunks that are approx 25mins each.
A standard harddrive will play back DVD material fine before you have to worry about harddrive speeds, but if you skimp on RAM, the OS will resort to dumping RAM on the harddrive in order to free up real RAM, and this dumping (disk swapping or page filing as it's also known) can be pretty slow stuff and will affect the rate at which you can read video (or anything else for that matter) thus hindering your real time previews.
If on the other hand you want to keep things simple and relatively fast you can do as I and others do and dump the videos to Huffyuv files. Instead of using the AVIsynth script to edit from, you just import it into virtualdub and save it as AVI using the Huffyuv codec. The advantage of this is that what you see is what you get, as in it's already processed and IVTC'ed and so no other processing other than playback of the file is needed (which can vary depending on the strength of the compression you use, again it's the diskspace/cpu time tradeoff)
That's great, but Huffyuv is huge lol. I've got the Ah My Goddess Movie on one of my harddrives and it's 62GB. As you can see you lose a hell of a lot of space, it's questionable really. Another thing to remember that Huffyuv is lossless and on old noisy source the files it can produce will be massive. It will eat your harddrive if you forget to remove the interlacing also.
The average datarate of the encode is 10.4MB/s (83061kbps), I hate to think what it peaks at, but if you have 3 or 4 of these as video layers while using a standard hard drive, you may start to struggle.
There are a few alternatives, I'm using a RAID array, which if you didn't know if basically combining multiple drives to form one big one, which results in increased speed and storage, you can check pretty much all you need to know about RAID through the links at the bottom of the post.
Again with how big/fast harddrive you need is down to how much RAM you have (read up) and how many video layers you will use. If you want to be a serious editor, want to do complex things or simply want an awesomly fast computer, I'd really reccomend RAID.
Graphics Card
You'd be forgiven for thinking that the graphics card is a key component in AMV making, I mean hey, it's all to do with video isn't it?
I was thinking about this the other day and concluded that as long as you have a graphics card capable of playing back the required resolution video you want to produce without bottlenecking the rest of the system, that it should be sufficient.
The only time I could see a nice graphics card being required is with any kind of 3D work, be it games, 3D modelling or something like CAD. Of course the more expensive graphics cards will have better hardware acceleration which in turn will relieve CPU stress and possibly lighten the load on the RAM a bit. You will also have access to higher refresh rates and resolutions. You can also expect general better quality, but the quality of the graphics card won't affect your final output, so that's worth bearing in mind.
That said, with a nice graphics card and monitor (preferably CRT to avoid dead pixels and shadows) you may have the quality available to you to fine tune your AMV, or see defects in your editing which you might not have seen on cheaper cards.
The Radeon 9800 card also used to have DivX acceleration, which I assume will work for XviD, unless it was a case of the software being optimised to make better use of the Radeon's features rather than it really being hardware tweaked for DivX
Links
RAID options explained
Harddrive speeds and what you should look for when buying a harddrive
The advantages of having RAID and it's real world results
Recommended RAID card manufacturers