Never view a prospective purchase in the dark
Check the engine and frame numbers against those on the V5 registration document. If they differ be very wary.
Check tyres, chain and sprockets and brake pads and discs for wear and amount of life left.
Check chassis bearings for excess play, notchiness, etc - eg. Headraces, front and rear wheel bearings, rear suspension linkages and shocker.
Check condition of plastics (especially the mounting and locating lugs, which can be prone to cracking and snapping off) and also the exhaust, frame, swing-arm and fuel tank for excess rust.
Try to start the engine from cold and listen for any unusual noises and look for excess (oil) smoke from the exhaust. Once warmed the engine should settle down to a constant idle speed.
It is always a good idea to listen/look at more than one example of a particular model to help get accustomed to what the really all do or don't as the case may be. For example GPz550s, GPX600s and GPZ600s can make a horrible knocking sound at very low revs, which is amplified by out of balance carbs - however it is not harmful and so long as it has disappeared by 1.5-2k revs, there should be no problem.
Specific things to look for…
GPX250R / GPZ250R / ZZ-R250
The liquid cooled, twin cylinder engine that powers all three models is usually very robust despite being able to rev to 14k. A good engine should be quiet, smooth and have a healthy power step from 9-14k revs. High mileage examples (50k plus) can wear out the top end - so check for excessive top end noise.
GPX and GPZ use 16" wheels and unusual sized tyres so if tread depth is 2mm or less, budget for replacements and very skittish handling. Also forks can twist very easily in spills.
Plastic on GPX is quite robust, however ZZ-R plastic is very expensive and GPZ very rare.
GPX400R
Liquid cooled 4 cylinder engine in a chassis based on the GPX600R. Some owners have experienced top end (valve) wear, listen for excessive ticking from the top end of the engine.
Original 2-2 exhausts are prone to rusting, GPX600 pipes will fit.
Check tail piece for cracks and snapped off mounting / locating lugs.
GPX600R
The liquid cooled, 4 cylinder engine is usually very tough.
The brake discs warp if the bike has been hard used and older examples may well require new headraces, fork oil and rear shock.
Pre 93 models have an electronic anti-dive fitted to the front forks, which may well have seized or have been simply disconnected.
Check tyres, 2mm or less will ruin the handling as will incorrect air pressure. Check locating / mounting lugs on the tail piece as they can crack and also check exhaust, which may well have been replaced with an aftermarket 4-1 system (Motad Nexxus is the most popular choice). If a race system has been fitted make sure the carbs have been re-jetted to suit, otherwise the power delivery will contain a huge flat spot.
GPX750R
Early models suffered from top end problems, later engines tougher.
Suspension components, tyres and tyre pressures need to be well maintained in order for the bike to handle well. Light bikes and worn tyres/suspension linkages do not make for happy cornering.
OE Kawasaki exhaust system prone to rust if you see an example with an original 4-2 exhaust check it carefully especially around the balance pipe.