After replacing battery, overheating and draining.

My iPhone 5 - 16GB now has like 2 years, battery was poor and had a great life, needed to change. Bought on ebay, changed, then charged up to 100% started to overheat and drained like 10% per min. Maybe faulty battery bought another one, same thing. Inserted old one, nothing like that, phone is cool, but battery is just 1000+ cycles and need to be replaced, i have to charge it twice per day. Can I buy here? Won`t be some kind of chinese fake? Tired of wasting money. Maybe its just phone`s fault?