Personally I would scan for any infections first, a fast draining battery can be a sign of such a thing, because the infection is probably running bluetooth searches constantly to find other devices to send to so they infect other devices, and just like any nasty it's aim is to spread itself as much as possible.
Although mobile infections are not "common" as such, they are out there, and they are growing, so the mobile world is going to be as big a target as PC's etc, bigger in some ways, so although it may be considered paranoia to run protection on your mobile, just like you would a PC, it's probably a good idea to start doing so, no point waiting for the horse to have bolted before you lock the stable door as they say.
There are quite a few options available in this field, and the names you would normally look to for your PC, also have mobile versions available, so the 1st thing I would do is install some mobile protection and run a full scan, see if that shows anything, it is far better that dismantling a phone, as thats the last thing you should be doing when nothing else helps, and this is free and simple to do, so can't hurt to try first, it "may" save a lot of messing about.
This doesn't mean there is nothing wrong with the hardware, but given the symptom it would be my first thing to do, if that shows nothing get Process Explorer from the App store, just like a PC a mobile has many things running in the background, there are many around, but I use Android Assistant, you should be able to see what is using your battery up, assuming it is a "software" issue, it also allows control on your battery, but always use such tools with caution, don't change to much in a single go unless you know what will happen.
Should none of this shown what is going on, then there is a high probability it is a hardware issue, then you can consider stripping the phone down, but if you suspect the connection terminal is weak\faulty, gently move the cable and see if it cuts in\out, mine does this, but it still charges OK, but I mostly notice it when connected to the PC, you only need touch the lead and it connects\disconnects like mad at times lol, so see if any of this helps before stripping your phone down, it could save you a lot of work, hope something here helps.