My iphone is completely broken, but I need pictures off of it?

I recently dropped my iPhone 5s on concrete and it shattered. It won't turn on and my computer won't recognize it. I've charged it fully, but still nothing. Right before I dropped it, I had taken some really great pictures from my vacation. I checked my icloud, but my pictures didn't transfer over to it. How can I get my photos? They're really important to me and I don't live anywhere close to an apple store.

If you saved the photos to the SD card, you can remove it from your phone and load it on your PC (you might need an adapter or a reader - see your computer store.)

Ask where ever you got the phone from to transfer it for you

Phone will have to have power for any of that to occur. You don't have an SD card. No iPhone does.
The problem is, any data on there that wasn't backed up to iCloud or iTunes prior might be lost in order to power on the phone. Very risky.
Only chance you have is to press and hold the power button and wake buttons simultaneously until the phone attempts to wake up.

Otherwise you'll have to restore it. Best luck.