I've done a few iPhone 5 screen replacements for friends/family- it's one of the easiest you can do.
Buy a kit like this for the exact model you have:
http://www.amazon.com/iPhone-LCD-Replacement-All-Networks/dp/B00KJL0A5U
You can probably find these cheaper, I remember finding for around $55.
I would recommend getting a complete kit like this from a reputable maker with lots of satisfied user reviews- I notice there's knock-off looking kits also, but I don't know as I'd trust them.
Watch the youtube video for the repair before you do it. Example:
https://www.youtube.com/watch?v=40Ph1J5ElpE
Watching it done a few times, you'll see what to expect- it's really bone simple. I could now do one in 2 or 3 minutes.
When finished, the iPhone should look and perform as good as the day it was bought.
There's ONE thing I found unexpectedly difficult with this fix- the ear piece speaker is actually a SOB to install correctly. If the little metal pins don't contact it just perfectly... it fails to work, and you'll have to open the phone all over again and reseat it. Front camera lens is a little tricky too, but not that difficult.
This of course assumes a kit that doesn't have all these elements assembled already: ie, you'll just replace everything, not use the speaker/front camera/cables from the original phone. The replacement kits I've used were just the screen and I needed to use the original parts. This maybe added a couple minutes to the process- and like I said, that damn earpiece speaker was the trickiest part. If you can, it may be worth it to just to get the fully preassembled kit.
But anyway, all-in-all a VERY easy fix.