iPhone 5 Shows Charging - But isn't

The short answer on how I'd do it: get both and test simultaneously. Although I have a feeling it is the battery, or neither.. since you mentioned a blue screen.

Here's the long answer: how I'd do it if I received this in my shop for repair.

First, since you mentioned a bad drop and a previous repair (by you), I would examine the logic board using good magnification. This is to detect drop damage and pry damage and try to rule them out, at least partially or the most obvious.

There are parts covered by metal shields, and parts that are not.

Those parts that are not covered contain important components that if damaged, may make your iPhone behave strangely.

So I'd start by looking next to the battery connector, next to the SIM tray (there is a glass IC/component near that area that if chipped will render your iPhone completely useless!!! So be very careful removing the interconnect cables that cover that chip), and around the screen connectors on top (I'd very carefully remove the "thin black foil" covering that area, since prying to remove it may pop tiny components off).

If I find something damaged, broken off, or completely missing, I replace it using micro soldering tools and a very steady hand. If you miss the prerequisites, post a god quality picture here and we'll tell you if it is a critical component and what to do next.

Second, I would test the phone using only a known good battery, lightning connector (you can overlay and connect those 2 while the logic board is still in it's case), and known good screen of course. Don't forget to disconnect the PMV (Power-Mute-Volume cable).

Once the setup is ready, connect the iPhone to your laptop. The iPhone should start (because the PMV cable is disconnected), charge, and get recognized by iTunes.

Let us know if something does not work as described.

However, if all is well, connect back the original lightning connector, then battery, to see if any of them brings about any bad behavior. Then the PMV.

Good luck.