The basic checks are to make sure all pages in the process work and are showing a valid secure certificate.
Check out for things that can stop people buying. Not knowing how much shipping will be, or finding out its expensive.Not being allowed to buy (happens a lot in Australia), not knowing if they can pay buy their prefered method. Being asked too much personal details...
Maybe have some freinds go through the purchase process and give you some honest feedback.
Watch them, but don't help. See where they stumble and how they get past it.
Afterwards ask them about the experience and what may have put them off when going through the process.
Give them some pants as a reward.
Maybe people find it hard to find the cart after adding to it.Maybe people don't like leabing yoursite for the chackout process and being forced to use PayPal.
I found out I could buy from Australia and shipping was free, but I had to go through several steps including giving my personal details to find out this crucial information. I would typically assume my country is not supported or shipping was extortionate and not bothered going that far.