I'll probably get shot down for it but DO YOUR OWN HPI CHECK. As in, the proper £20 job, not the £10 basic check, nor any other companies offering a similar thing but cheaper. In addition to the actual check, there's a guarantee that if the check doesn't reveal an anomoly (for example, log book finance owing which doesn't show up) they'll cover your loss. It will appear poor value for money on a cheap car - but these are the ones most likely to have something to find. And for an expensive car, you have more to lose.
Also - might sound counterintuitive - don't dismiss a car with a "high" Auto Trader price marker. The reason is, dealers know how much their car is worth and with the internet age, it is very rare to find a dealer just adding a bit onto their cars in a hope. The high price marker will be for a reason, for example it has certain desirable (but not picked up by the price marker) options or colour; or its simply a great car which has led a trouble-free life. Obviously, its a handy feature for buyers but understand why it is what it is. This also applies to cars with a "low" marker - in the new normal of mostly doing the transaction online, it could very well be that the condition/bodywork is generally poor (it is very difficult to accurately gauge bodywork/paintwork condition over the internet, and very easy to choose not to photograph imperfections!)