iPhone charging issue

My iPhone is having issues when I plug it in to charge. Sometimes it goes from charging to not charging and back and forth and won't charge the phone. When I plug it into the wall adapter it displays the charging screen and then immediately displays my wallpaper and goes back and forth as if I'm plugging and unplugging the cable. I've tried multiple wall chargers and cable combinations and still have this problem. It also happens sometimes when I plug it into my computer.
Any thoughts on how to fix this?

Another issue is that my phone won't charge when I'm talking on it. For example, I'll start with the battery in green and then I'll get a 20% battery life warning and sometimes I'll get the 10% battery life warning and then my phone will just die (spinning icon followed by shut down). I then have to wait for it to restart and I get a few minutes of talk time (enough to tell the other person my phone died) before it shuts down again.
So my question is this: is the iPhone supposed to be able to charge while talking or is there something wrong with mine? I've gotten it replaced at the apple store before (my silence slider fell off) and both new and old phones wouldn't charge while talking.

Thanks for the help