Hiya Danny
If i were buying second hand, here's what i'd be looking at...
1. Spin the wheels, are they buckled or untrue ? A little wobble's probably a quick tweek up for a bikeshop.
Do they spin freely ? Let them spin to a stop, (making sure the brake blocks aren't touching) any sudden stopping or graunchiness indicates hub bearings need attention.
Stand over the bike, finger and thumb round each tyre and feel if there's sideways play, again indicates the hubs need attention.
2. Apply the front brakes and 'push' the bike forward. Any excess movement between the frame and forks , it incicates the headset requires attention.
3. A quick look round the frame, particually at the points where its welded...particually all round the BB, where the seat tube joins the crossbar, at all the joints, look for cracks. Very rare, but it does happen.
4. Look at the front chainset, are the sprockets excessively worn ? Look for a sharkfin effect. If they are, it probably means the chain and rear cassette are too. Worn transmission like this costs a fair bit to put right.
5. Preferably (certainly) test ride the bike, change through all the gears front and rear. TBF, if its difficult to select an odd gear, its probably just an adjustment thats required, but if they start skipping and over-running etc...it indicates worn chain, worn cassette etc,,,again, expensive to put right.
6. Tyres...are they ok ? Its easy to forget how expensive tyres are...£30 for a decent pair.
7. Wobble the cranks...any excess play ?
Don't be put off, there's plenty of hardly used bikes out there, brought by well meaning people who just don't get to use them much...but a wrongun will cost you.