I would always have a good look round a car (including getting down on hands & knees for a look under) and in engine bay, under boot carpet etc, to look for any signs of damage. Take a mental note of all damage, and consider the cost for replacement / repair. If car is being sold as 'mint' then I would argue the price down based on any imperfections. If it is a well used car, and this is reflected in the price, then I would only use things like a bald tyre or dodgy exhaust as discussion points.
The amount of price reduction I would expect on any car would be based on how reasonable the asking price is in the first place. On a £20K car I would aim for £1K off, but would prob be happy with £750 off. On a £1K car, I would aim for £250/£200 off, and would be happy with £150ish off.
I would not kick tyres

. But would look at tread wear, across full width of tyre (as this not only shows how good tyres are, but also if tracking is out etc). Look all around for accident damage, including boot floor & under bonnet. I am always suspicious of an older car with a very clean engine bay.
I would research what something is worth, and point out what other similar cars can be bought for. Point out what is needing spent on a car, to get it up to the described condition/ value. Bear in mind, if someone does not sell a car quickly, they may need to spend money on repairs / tyres etc, so that is a worth while negotiation point.
When buying privatley I always judge the seller, and their home as much as I judge their car. To some extent you can do the same with smaller independant dealers. It's always good to establish why they are selling a car.
When negotiating, don't budge to quickly on your lowest offer. Silence is always a very powerfull tool. If you have a figure in mind, (for example for a £20K car, 19K), it's worth saying, I only have a budget of 19K, we'll do a deal now for that?, then wait for their response. If they come in at 19500, then don't jump in, pause, see if they lower it... if they use the silent treatment on you, then repeat your offer. Wait for their response... it is never good for things to go sour, be friendly and pleasant. At that point it may be worth giving a little, and offering 19100, along with a comment like, that is really pushing my budget, I do like the car, but can't afford to go any higher. If they won't budge, then question how low they can go. If they still won't budge, then be prepared to walk away, leaving them with your number. Unless it is already a good price for the car, and they have only just advertised it, and had several other enquries, then I would walk away, and give it a couple of days. IMHO
With a dealer, it's worth talking the price down, then adding as much as you can; road tax, tank of fuel, mats etc to seal the deal.
I sold my elise a few years ago, when it was only a few hundred miles from it's next service, had 1 week's MOT & Tax left, but the buyers never asked, so I did not point these things out. Always ask for the above! I was aware they may ask, to just kept talking (enthusing about the car), and kept them of the subject.
