I’d check for :
Play in the suspension bearings/bushings:
Grab the bike by the top tube/seat tube junction and lift gently. If you feel a knocking, the shock bushes are probably shot.
Grab the rear dropout in your other hand and wiggle it from side to side. Again, if there’s play, the suspension bearings may be shot.
Blown fork and shock seals:
Have a good bounce around on it. If there’s an excessive amount of oil (or black streaking) on the fork stanchions or shock shaft, then the seals may have had it.
Drivetrain:
Ideally, take a chain-checker tool as it’s hard to check this by eye or feel. If the chain is over-worn then you may have to budget for new rings and cassette.
Wheel bearings:
Grab the wheel rim and wiggle from side to side. If there’s a lot of play, wheel bearings might be knackered (check QRs/maxles are tight though!).
Brakes are easy enough to check by lever feel (and function!). Then have a good look all over for cracks. Swingarm and bottom-bracket area seem to most common, but could be anywhere / on any weld really.
None of these are necessarily deal-breakers on their own (except cracks!), just means you have to budget for new bits or haggle the price down!
Shock bushings are only a couple of quid, but if you don’t already have the tools/know-how to replace, you’re probably looking at £20.
Suspension bearings anywhere from £15 to £80 depending how many there are.
Fork seals about £25, plus service cost if you can’t do it yourself.
Shock service about £75-£100, around £25 in parts to do it yourself.
Drivetrain probably £100 for cassette, rings & chain.
Wheel bearings about £15 front, £30 rear.