How to backup my iphone 5c if i can't get inside my phone?

My i phone 5c screen is broken and the touch is also messed up. I have ordered a new phone and it should arrive tomorrow. The thing is, i have alot of data on apps such as my music since i don't use itunes. Will all of that data transfer over to the new phone once i switch the sims or log into icloud or will i have to back it up to icloud. I don't know how i can back it up since i can't get into my phone.

This is why you do backups regularly. This could be a good learning experience for you because you will now probably lose everything on it.

Actually, you should develop a good habot to make iCloud and iTunes backup regularly. Then you can recover your data easily.
But there's still one way to transfer your data to your new iPhone.
Firstly, do not add new data to yuor new iPhone, because if you transfer old data to your new iPhone, then your current data on your new iPhone will be erased.
Secondly, get a recovery tool from internet. There are many such tools. I have used fonelab for ios, and it works well by now. Connect your old iPhone to your computer meanwhile, launch the tool. And then do as the guidance in that tool to continue your next steps. That's the easiest way I have taken. You can have a try by your self.
Hope it can help you.