I am new to this forum, just reading your post and thought id point you in the right direction.
The 6 series is a tricky car to buy - get a bad one and your credit card will be melting, get a good one and you will have years of fantastic driving in front of you.
Firstly, IMHO get a leter model as you can, the early cars had a different variation of the 3.5 engine which wasnt as relaible and far more costly to put right, newer lump fairly bulletproof.
Your looking at an 88 car so it will have the good engine. Most common problem on the 6 is rust, if not looked after it will rust away around you. Places to check are the front wings, these are the most common place and are pricey to replace, in the UK a new wing is around £4-500 - thats part only! Most 6's will have probably had these done by now, so check the quality of the work. If they are rusty then make sure you have the money to repair them - properly! The wings will rot all over - check EVERYWHERE!
Next weak spot is the rear arches, these again rust badly. Rust also is common around the rear number plate as water gathered there and had no way of escaping, not to bad to put right. Rust around the boot lid, under the seals again a common fault.
Check ALL seals! They perish with time and are costly if bought from BMW. ALso check the sills for welding on both sides - not a problem if its been done right. Front sub frame is weak - may need welding to - check!
Mechanically, well the engine should have full history, its a chain cam so no need for a change until around high 100k upwards. Suspension/steering components can often need replacing, not to pricey but it can all add up. Exhaust not cheap, bumpers not cheap! - infact not many parts are cheap! Alot of the mechanicals were shared with the 535i so easy to source. Switchable sports auto best box, manual - no thanks, leave that for the M635i
Ideally you want a Highline, this was a ltd edition and was on late cars only. Highlines came with nicer bumpers, and so much leather it had cows running for miles when it drove past! Dash, seats, roof, pillars, center console, doors, sun visors - pretty much the entire interior is leather - and hand stitched, the workmanship is A1 - I have a Highline and you have to pay A LOT of money these days to get workmanship like this. Also try for the electric Recaro sports seats, these are superb. Make sure all the electrics work, if the headrests dont go up and down then dont worry as its easy to put right. If the seats feel wobbly, again - easy to fix! AC is nice, not many have working systems these days as compresors are expensive and regassing isnt worth it on a car that gets used mostly as a weekend toy.
Ideally you want the original metric wheels - these have tyres which are ultra rare to source these days as they are 16.5" (converted) Makes an orginal car though, a lot of owners have replaced them with normal wheels from other BMWs or Alpina's - I have the originals.
Also nice to get the de-chromed version, ultra rare but looks far nicer IMHO. If the car looks tampered with walk away - as I said - these are costly to put right! Not sure where you are in thw world, but a good 635 in the UK on a late plate is about £4-7000, dependant on condition/miles/owners etc
Its a false economy to buy a cheap one - it will cost you dearly getting it back to good condition.
LSD is also a rare thing on 6s here in the UK - great to have! Good luck in your hunt - but remeber to not buy the first one you see! These are easy cars to fall in love with, hence make mistakes with so be warned!
I have had mine for a while now, and I will never sell it as long as I can drive. Compared to modern BMWs its so visually stunning and the drive is superb for such an old car.
Hope that helps!!!