The cheapest and reliable way to unlock iphone 5?
The reliable way is to get the carrier to unlock it. In the US - this means


Verizon's are sold unlocked in the US whether you have a contract or not. 
AT&amp;T's are locked, but they have a process to get it unlocked once your contract is fulfilled. 
Sprint's situation is unclear.
T-Mobile's are reported to be sold unlocked in 2013 when they start selling iPhone.


Cheapest depends heavily on your specific contract details and if you want to risk paying a third party to do something the cell carriers do not like. You could always look for a jailbreak and circumvent the lock, but between paying off your contract early or selling your locked phone and buying one that is legitimately unlocked is an easy, and simply about time and money now that unlocked iPhones are mainstream in the US. I suggest dealing locally in selling and buying an iPhone - connect on craigslist or wherever and meet in an Apple store so that you can check the serial number - be safe in a public space, etc...