iPhone 5 not charging when cable is connected

I recently had my iPhone battery replaced, it was ordered directly from the iStore (I live in South Africa) and is a valid, not generic, battery.

Since the technicians replacement, the phone has been working like normal, but upon charging for the first time I noticed the battery percentage would not improve.

I'm really confused about this - and frankly, quite frustrated that when connected to any power outlet, the phone recognises that it is plugged in by displaying the lighning bolt next to the battery life bar, and makes the "charging sound". When I tried to charge it through my computers USB port, it does the same - displays the lightning bolt and makes the familiar sound. My iTunes even recognises the device and if I wanted to, I could have synched my music.

My previous issue before having my phones battery replaced was also charging related - since updating to iOS7 in September 2013, the battery life was absolutely dismal, I could take a photo on 100% battery, and it would die almost immediately afterwards. To use the phone effectively I had to keep it permanently on charge.

My main concern here is that since having the aforementioned problem fixed by buying a new battery, it will now not charge at all, unlike previously where it would actually just charge but die almost immediately. I have scoured as many forums online related to my problem, and I have tried everything I possibly can to get my phone to charge. Unfortunately in the time it has taken me to sort it out, my phone has died and now will not switch on, even while "on charge."

Coukd this be a problem with the phone itself?
Should I have this seen to? I am beginning to feel quite disillusioned with my endless phone problems.

Sidenote: My iPhone was a gift from my father who bought it in England for me while on holiday in November 2012. My sister has one too - and I have tried charging my phone with her cable - no result. I don't think the warranty will be able to help at all, I doubt my dad still has the papers.

Please can someone shed some light on what I can do, or if anyone has had similar problems?

Thanks in advance