Does not charge or recognized by pc

Hi there, besides my Ipod trouble I also have an Iphone issue. Since there are 4 in my household I suppose it was bound to happen:) Anyhow one day my Iphone started to turn itself off without any reason at all turned it back on and this happened multiple times. Now the Iphone can not be charged by the Computer and is not recognized. Stays absolutely dark and dead when plugged in the PC on any USB port. When I plug it into my AC adapter it comes on and charges but the moment I unplug the cord from the charger the Iphone dies again. Anyone have any idea. New Battery? New Logicboard? Iphone4? Thanks.