Most guys have most of these tools lying around, however some guys had to discover that they need a lot more tools to get the job done. You can get by without most of these things; I just happen to believe that a little overkill in the tool department is never a bad thing.
Parts kit preparation
A rotary tool - I recommend a Roto-Zip, but you can get by with a dremel. The RZ spins much faster, has more power (so you don't have to break contact every few seconds to regain speed), and will accept Dremel bits. The RZ-20 package came with the Zipmate right-angle attachment and a metal-grinding disc, which is needed for the rivets. Any Roto-Zip will work for this, as long as you get the Zipmate with it. If you don't get a Roto-Zip, I would recommend buying a stand-alone right angle grinder.
Two pairs of pliers. I recommend one set of vise grips and one nice long pair of Channel-Lok pliers, as they keep your friend's fingers far away from the cutoff wheel. You need two because you may end up having to bend off bits of the receiver, however if you're grinding the rivet heads properly, you probably shouldn't need to pry. If you're doing this alone, you can get by with just a vise and one pair of pliers.
A fairly long leather (or other tough, heat-resistant material) glove. I personally wore a welding glove on my left hand, as it was getting showered with sparks from grinding; no glove on the right, as it was shielded from sparks and I feel it's safer to hold the rotary tool with a bare hand. After demil'ing four parts kits and constructing one bolt cutter rivet tool, I noticed that the glove was literally caked with metal filings which didn't wind up on my hand. This is a good thing.
Eye and ear protection. With little bits of metal and potentially broken parts of cutoff wheels flying around, eye protection is mandatory. I wear ear protection as well, but that's just a comfort thing. Generally, I use the ear protection only when I'm running the roto-zip, which is quite loud.
A respiratory protection mask. When working with metal, it's not a bad idea to keep your lungs and throat clear of metal filings. It's probably not a huge deal if you're only doing one parts kit, but if you build any more, you really ought to wear the mask whenever you're grinding. The wimps among us may find it appealing that the mask also keeps sparks away from your nose and mouth.
Center punch - for the rivets, to create guide holes for the drill press.
Set of standard steel punches - this is for punching through the rivet holes.
Pointy punches - not sure what they're called, but the kind you use to create pointy indentations in metal. This is used to start holes to guide the drill.
3lb hammer - You'll be doing a lot of pounding, and a small hammer just won't cut it with the rivets.
Drill press - again, for the rivets.
Needle files - you might use them to expand the rivet holes in your receiver or for fitting the magazine well.
Optional: a gear puller. A 5-ton gear puller with 6" jaws can supposedly pull a barrel out of a trunion without requiring that the receiver be cut apart first, even if the rear trunion is installed. This can be useful when backing out a barrel press gone wrong. Forums report that the Craftsman puller breaks easily, but the Auto Zone one works well.
Ways NOT to Build an AK-47
Pop rivets, or "Home Depot rivets": These rivets are nothing like the kind of rivets you need in order to build an AK. As opposed to hollow rivets (generally aluminum, although steel does exist as well) expanded by being yanked through a rivet gun, the rivets used in an AK-47 start off looking more like unthreaded screws, and are crushed under literally tons of force until the metal flows uniformly around both sides of the hole.
Tools needed for assembly (rivet build)
12-ton press. They're useful things to have around; AK's aren't the only thing you're likely to use this for. I've read that you only really need a 10-ton press, but the 12-ton one was only a little bit more. If you're going to do a lot of AK's, then I recommend a 20 ton press. I've found that there are some barrel pins that just flat-out do not want to move for a 12-ton press.
Several jigs. These are pieces of metal shaped to keep your rivets in place while they're being crushed together. More on this later.
A pair of 24" bolt cutters to turn into a rivet crushing tool, if you can't afford the rivet tool below. Expect to spend several hours getting it just right. You'll need a right angle grinder or bench grinder for this part. You can use a 18" pair of bolt cutters, but the 24" is more versatile; it can crush rivets on underfolder kits, whereas the 18" can't.
AK Tool Pricing
I ended up spending a considerable amount of money getting a good collection of rivetting jigs together. Ultimately, I managed to avoid re-buying a lot of parts, but there was some overlap. My recommended tooling set is: