While a Carfax report is generally a good idea, it may or may not be accurate. Carfax checks with the DMV, some insurance companies and maybe a few other entities. If the car was involved in a collision it may have not been reported to the insurance company if the repairs were paid by the owner, therefore Carfax would never be aware of it.
My advice is to ask the seller if you can take the car to an automotive business (independent mechanic or another car dealer) for an inspection. These typically cost around $50. If you know a reputable mechanic, bring it to him/her. If not, any major dealer will be happy to take your money. For instance, if you're buying a Honda Civic from the Honda dealer, take the car to the other Honda dealer across town, or in the next city (provided it's within a reasonable distance), or to the Toyota dealership provided they're not owned by the same person/company.
If the seller says yes go ahead and have the car inspected, you can feel safe that the seller has nothing to hide. If they say no, walk away. Do not try to negotiate, or anything, just walk away.
Second piece of advice. Don't buy on the same day you look/test drive the car. Look at the car and test drive it on say Sunday, then take a day or two to think about it. Look/testdrive again if you feel like it. Finally if you decide that this is the car you want, then go and negotiate.
Third piece of advice is to know the average price of the car you intend to buy. Use edmunds.com and autotrader.com to get a price range for the car you're interested in (Compare cars with similar options and mileage).
Fourth piece of advice. Assuming you're buying from a dealer, buy during the middle of the week at the end of the month. Dealers are typically trying to unload as many cars as they can during month end and the middle of the week is the slowest time (as opposed to the weekends). Eat an early dinner then come into the dealership at around 4:30PM - 5:00PM. Spend all evening dealing on the car if necessary. He/she probably have will not have had a chance to eat, and will be eager to complete a sale and and get the "hell out of there". Car salesmen are human too.
Fifth piece of advice. I understand you've already been approved for a loan by your bank, but sometimes the dealer will "shave a little off the price" if you finance through them. You can always refinance with your bank at a later time. Also, negotiate the price of the car, not the payments.
Last piece of advice. If you're not satified with the price offered by the dealer, leave. If you're offer was reasonable, the dealership will likely call you back within a day or two to see if you're still interested. If they do, round 2 begins.
Good luck.