How to transfer my data from my old iPhone to a new one?

Okay, so my old iPhone 5 is broken and won't connect to my computer. I'm fairly sure I backed it up on my computer, but I don't know how to access all the information. If I were to get an iPhone 5s, would it be possible to transfer the backed-up data from my old iPhone 5 to my new iPhone 5s? I would just go get it fixed, but since Apple's repair prices are rapage, I did it myself, so now they won't even touch my phone. And does the iPhone 5 and iPhone 5s share the same SIM card? So i can just move it to the other without having to go through AT&T to get a new number?
Thanks in advance.

Added (1). Or should I just try to fix it?

Added (2). My iPhone can't turn on. The power button is broke, and it will not charge. It will not connect to my computer or any power source in any size, shape, or form.

Added (3). Okay, I found my back-ups, but hey're encrypted. I don't know how to get them un-encrypted. I also don't know how to transfer the data over. How can I "transfer the data to my pc, then to my phone"?

Okay if you still have full access to the 5 go to settings icloud and backup now, then when you get the 5s during setup you should sign in with your apple id and it should work. If you don't no worry every time you charge your iPhone in sleep mode overnight it automatically backs up so just sign in and back up on the 5s and you should be fine.

Its not there.

1. Yes, iPhone 5 and iPhone 5s share the same SIM card.
2. But iPhone 5 backup file is not works for iPhone 5s. You may try this data recovery for iOS, it can read your backup file. So you can export the contacts, songs, videos, photos and apps from the backup file and resync them to your iPhone 5s.
http://www.reviewstown.com/data-recovery-for-ios.html

You only can do to trasfer your data to your pc and the to your new phone