This is exactly what happened to my 5c. After upgrading on Sept. 23 it would not recognize ANY Apple brand charger, even at the ATT store where I bought it. Once the battery drained, it was completely unusable, leaving me no way to retrieve the serial number needed to contact Apple. ATT said to replace the battery and recommended a local (non-authorized) repair shop since the nearest authorized dealer is 45 miles away.
New battery didn't help - charging still not recognized - so instead they put on a new charging dock (with my old battery back in.) It cost me $70 and it now works!
After reading similar posts, I believe the ios9 update sent a power spike that either killed or disabled the dock or some related ability for the phone to recognize that a charger was attached. I called Apple to request they reimburse the $70, but the senior tech was adamant that the software update wouldn't affect battery or charging . . . and even so, my trip to the non-authorized shop voided warranty. I explained that many others are having this problem, so even though he says it's not possible, SOMETHING is going on. I asked "wasn't this update supposed to improve battery life" but he wouldn't go down that road.
Finally, I told him I can't drive 45 miles on a gamble, and can't wait for Apple to admit a problem - so having it fixed locally for a reasonable price was my only choice to get a working phone. Again, he just stood firm that this was my fault. I just want my $70 back. Hopefully Apple will reconsider as more people report this same problem.
As a side note . . . I also have an iPad mini which I also updated around the same time. It charges, but now will give an occasional alert that I am using a non authorized charger, which is not true. Prior to my phone dying completely, it also up the warning even though I only own the two AUTHORIZED chargers - one for the ipad and one for the phone. I also work for a school district and have access to numerous authorized chargers purchased with brand new iPads. Nothing worked until I replaced the charging dock.