The issue may lie with either the dock connector or the battery. Either or both items may need to be replaced. The cheaper solution would be to change the dock connector.

What are the specifications on your laptop? Everymac has some information but they span between the first generation and second generation. http://everymac.com/ultimate-mac-lookup/?search_keywords=a1304

Have you tried the following steps listed here: http://support.apple.com/kb/HT5943

Has the phone been damaged in any way (i.e. water or physical damage)? If not, try hard resetting the phone (hold the power and home button for 10+ seconds while connected to power). If that doesn't work, you may need to replace either the charge port or battery.

Replacement digitizer/LCD for iPhone 5s has gone down in price. They can be found for around $200 for the set if you know where to look.