How to backup contact in iphone is the touch screen is broken?

I dropped my phone 5c and now the touch screen doesn't work at all.
I bought a new iphone and I really need the contact information in my 5c.
It can be connected to itunes on my laptop, but I didn't find any "contact" info.
I tried backup up once, but failed.

So, are there any there ways to get my contact info? And put that into my new iphone directly?
Can I control my phone though lap top?

ITunes will back it up - if you can connect it to the PC/Mac and power it up
you should not need touch screen, as I recall, but if there's a screen lock, you might have to put the pass code into iTiunes