Sarah, if there have not been any other issues with the iPhone. i.e. water damage, getting dropped etc. I would start by cleaning the dock connector. Use some isopropyl alcohol and a few puffs of compressed air. Also, make sure the pins are not corroded, bend or broken. Then replace the battery, it does sound like a battery issue. Also, with a new battery, you will have a known good starting point for further trouble shooting, should that be necessary. Hope this helps, good luck.