Most people buy used cars with their eyes and based on how much they have to shell out of their pocket. Nothing more nothing less.
Of all the used cars I have helped friends/family buy I have always looked underneath, looked at the brakes, body, tested all the accessories for proper operation, stress tested the battery, checked the fluid colours, looked under the oil cap with my boroscope, into the cylinders (where possible), taken it for a test drive and ALWAYS ALWAYS ALWAYS hook up a professional level scan tool and scan for codes not just the generic P-Codes but for OE specific codes both Stored and Pending. Also look for at the very least if the Catalyst monitors are not ready. This is a dead giveaway that they have most likely just reset all the codes to turn off the light before you looked at it or that there is emission related components not functioning properly.
This is above and beyond checking it for past accidents registered against the title and looking for signs when you see underneath it where cuts/welds may have been done. I also try to see if the engine / transmission look like they were replaced. Eg: Shiny metal around bolts / eletrical ground that appear they have been moved or a dirty engine attached to a shiny clean rebuilt transmission type things. Sometimes you can match up the VIN#s or partial VIN#'s stamped on the engine to the actual VIN on the vehicle itself.
And as always ask for receipts, service records, maintenance logs or any other records they might have, as these are a great thing to add to the evidence if this is a good purchase or not.
If you can't do the inspection yourself, get a mechanic to look at it. It's worth paying $100 to have it looked over and having to walk away then paying thousands because some major problem was hiding.
There are a ton of "I'll sell it rather than do proper maintenance" folks out there.