sounds like a ‘part time’ trader. someone doing a bit on the side. personally i’d avoid unless the car was cheap enough. he’ll be taking a tickle out of it with no intention of taking any responsibility should things go awry.
to answer your original question, you just rock up at the PO with your insurance cert for the car, valid MOT cert, and the completed V5, and they’ll let you tax it and send off the V5 at the same time. there might be another form to fill in, but if so, nothing major. however, make sure you get a receipt from this guy, see some photo ID confirming his address ( which you should also note ), and get it dated and signed.
you don’t want to end up liable for the period of time the vehicle has been untaxed, and for all you know it could have been through every speed camera in the land at mach 2….