Absolute cheapest, a couple of daylight bulbs in inspection lamps, muslin over the front and a wall painted matt white! We are talking about £40 of odds and ends!
Slightly more seriously, I would recommend looking at the cheap Chinese lamps on eBay, some of the slightly higher power ones are surprisingly decent with only downsides things like slow recycle and very plastic build but totally usable. I would recommend buying at least 2, stands, softboxes of at least 30" and then some kind of trigger, you can use either wireless or a cable. A cable will be cheaper but the wireless system isn't much more and far less hassle. Most flashes trigger from each other so you don't need any complex cabling, just the ability to trigger one strobe. Only issue with that is obviously you can't easily use them in-situ with another flash system but that will probably not be an issue!
One strobe (in a similar size box as mentioned above:
Two strobes:
You can start to get more complex with three+ strobes which I have done before but tbh it is more hassle than its worth and you can get very good results with just two cheap strobes.
Oh and with regards to a backdrop - for those two shots I used my projector screen I use with my home cinema but again there are pretty cheap options you can source from eBay
Non-reflective and thick backed is what you are looking for.
