Which printer has the cheapest ink

its a nearly impossible question to answer, different manufacturers put different amounts of ink in their cartridges, use different amounts while printing, use different amounts for print head maintenance. They often have "page yield" ratings, like a cartridge can be expected to last "500 prints" but each manufacturer may use a different print to test this. And its continuous printing not including maintenance cycles. Then there is tri-colour cartridges, vs separate colour inks, and printers with more than 3 colour inks. I would say stick to a printer with 3 separate colour cartridges, there are some that use between 6 and 8 cartridges and will be expensive. If you want something really cheap to run, get a laser printer.
 
You can get the ink refills that come as a bottle of ink that you refill your existing cartrige with. This will end up cheaper then buyoing cartiges all the time.
I have found that HP inkjet printers can get a lot of coupons from HP for getting money off your next ink purchase. They will email you the coupons and stores like Office Max and BJ's wholesale will have coupons for HP ink cartridges.