I have this same problem with my 4s. (white 64 gb, factory unlocked)
I was charging it one night, and when I went to disconnect it was extremely hot to the touch. Next morning I went to turn it on, and - nothing. Plugged it into the wall charger, got it to turn on, but when I disconnected it - nothing.
Plugged it into my Macbook, and got the same Apple logo on and off as the OP said, but never went on completely. When I tried the wall charger again, I was able to re-set everything and get it to the point where it told me to connect to iTunes. When I connected to the Macbook again - nothing. Only the white Apple logo going on and off.
I did eventually get the phone into recovery mode whilst connected to iTunes. It said that it detected a device in recovery mode, but eventually came up with an error message (forget number) that said it was unable to recover the phone.
Since that time, the phone is basically dead. I had lots of problems with this phone previously. I bought it in the US (unlocked for GSM) for use here in Thailand where I live. I had nothing but problems with the phone part of it from day one. Constantly dropped calls, losing signals, running hot and draining the battery looking for a cell signal. It's been an $850 nightmare for me. I've been to the Apple support forums, and seen others had same or similar problems, but no resolutions. And no acknowledgement whatsoever from Apple that there is or may be a 'known' issue(s) or glitch(es) with this phone (or particular model). I've also written to Apple asking for advice as to what my options might be, and received nothing in the way of reply from them. Very frustrating...
And in Thailand there are no 'official' Apple shops. They sell legit iPhones here, but trying to get service by Apple is non-existent. Having given up all hope of a happy resolution, I'm off today to a local shop that (supposedly) specializes in Apple/Mac repairs. Of course this will void my warranty (phone is less than 7 months old), but at this point I'm stuck and somewhat desperate.
I have a feeling that the battery is fried, but that's just a hunch on my part. If I can get it repaired to a workable condition, I'm going to cut my losses and sell it. I'm quite disgusted and disappointed with Apple about this.