You will need to remove the barrel by pressing out the barrel cross pin. So you will need an AK Barrel Pin Tool from Gunthings.
What size ram does your press have?
And when you get the pin out you will need something to press out the barrel. I would suggest using a 3/8 grade 8 bolt about 5" long (cheep & it works) with a steel nut and a brass nut to protect the barrel.
You will need a trigger guard rivet jig if you are working from a shop you could make one of these.
This will do everything the AK builder jig will do for a whole lot less, apparently. $255 for the rivet jig, long rivet jig and trigger guard jig.
As always, there are ways to work around the cost... but the jigs make life a lot easier.
$10 bolt cutter from harbor freight and about 2-3 hrs of elbow grease. You can modify vise jaws for a trigger guard jig if you have a drill and some punches for next to nothing (if you already have the vise) just time involved. The barrel pin, 9 times out of 10 can be knocked out with a hammer, punch and a solid base. etc etc.
Even better if you can find somebody near you that already has all that stuff and can help out.
I checked out the original plinker. That looks like a good deal. I have money but building ak's isn't a huge priority for me. Just want to try it out. But will spend the extra money for the tools to do it right rather than cheap out and be less than pleased with the results. I probably wouldn't have gone out of my way to buy a press for just ak building and the occasional wheel bearing but now that I now that I can get access to one it makes the process a bit more appealing.