How to fix not charging phone?

Hello,

i have some problem with my iPhone 3gs 16 gb. The problem occured a few months ago when phone randomly shuts down even when battery was charged at 50%. Somehow i manage this but recently phone has died completly. The battery drained out and connecting it to a charger haven't affect phone a bit. So i figure - this must be battery. I bought a new one, but after i have mounted it phone turn on for a few seconds and again - dead. I thought that maybe there is something wrong with a new battery. I charged new battery using two cables, + and - from usb cable which i connected to VCC and GND on battery. I have charged it almost to 80 %. The phone was jailbrokem so i restore a clean 6.1.3. Well this doesn't help at all. When i connect phone to a charger phone detect it, and charging icon appears. The problem is that battery is not charging, but draining very fast. So after few minutes phone is completly discharged, and i cannot turn it on again. Also iPhone is getting very hot when connected to charger. This trick with ring/vibrating button and power/home button doesn't help. Do you have any idea how to fix it? It is not a battery, charger or dock connector failure. It must be logic board issue, but i'm not quite sure how to fix it and why such problem occurs.