I won’t offer because I tend to buy bikes every decade or something and normally nearly new from a dealer.
But… Start by going down to a main dealer (if poss) and inspect and sit on the same bike/model that you want to buy. Ask the dealer about the marque and mention you might be interested in buying an older version (if relevant) and see if he’s got any tales to tell. Do you’re homework before setting out. Know the service interval and any model peculiarities and recalls.
However, if I was with you on either Saturday the 2nd of Feb or Sunday the 3rd ;), I would ask for all the paperwork before even looking at the bike. I would check the frame and engine numbers against the log book. If anything been has filed or tampered – it’s stolen and you just walk away.
Next, obviously, service record. Make sure the book’s stamped up by main dealers and the service interval has been observed. Whether you accept the “I do some/all my own servicing mate” depends on what job the person’s got but that’s another point at which I would walk.
Check the service receipts carefully. This can be a give-away as well, as sometimes you see things being done that aren’t exactly service items: New fairing, handlebars exhaust system!
Ask the usual: how long s/he’s owned it, what type of riding they do, any crashes (most people are honest). If it’s a lot of money you could do the HPI check. It’s probably worth it for a bike since a heavily crashed bike is never the same again. And anyway, the person you eventually sell it on to might do an HPI.
The mileage and the riding type ought to tie up. Long distance commuting will rack up the miles but a high mileage on a fair weather only bike means someone isn’t telling the truth or most of the miles came from the previous owner to him/her. (An unknown quantity.)
Ask why s/he’s selling it but it’s a non-question and one for which I’ve never had the skills to assess the possible answers.
Look and feel for burrs on the extremities: handlebars, levers, footrests, under the exhaust (I dunno what bike it is so this might not all be relevant). If s/he’s had a spill that they didn’t mention at the start I would start to be suspicious but use it as a haggling point if it’s cheap. Small spills on a dirt bike are expected.
Look for general tattiness. When you sit on it, does it sag as much as the one you sat on in the show-room? If so maybe the suspension is on the way out. Maybe ask about suspension settings (is the owner a tinkerer? Might even have got everything on stupid settings).
Get a test ride. Listen to the engine and feel the clutch. Try to feel whether it’s wanting to pull left or right all the time (bent frame). Look out for a smokey exhaust. Normally very bad news. Look for oily dirt (small oil leak) or patches of bright metal – big oil leak?
If you’re still happy, think about the owner. Basically, I try to build a profile of whoever is selling the thing. If the bike looks OK and I really trust the guy, I’m happy. If s/he’s a bullshitter, I would look elsewhere.
So be prepared to walk away, never think you’ll not get another chance unless you know it’s for you. Then have the readies and be prepared to haggle and give the impression that you’ll walk away. But say I’ll give you the cash now if you’ll take this price. Whether you walk or not depends (obviously) on whether it’s a bargain or not. Knowing this depends on whether you’ve done your homework.
Don’t be phased by other people turning up and expressing an interest as this can be a ruse to get the unwary to think there’s more demand than there really is. Mobile phones going off with a “Yes, she’s still for sale…” is enough for me to walk.
[And for goodness don’t buy off anyone who has a web page with a link to an XHTML checker that fails! I mean close those XML tags. – only visitors to a certain web site will understand this.]