STEP 1) Sign Up For An Account At GitHub
STEP 2) Find Repository Of Cryptocurrency You Wish To Fork
For this example we will use Bitcoin Core
STEP 3) Click Fork To Create Your Own Forked Version Of The Repository
STEP 4) Edit Branding And Network Rules (Advanced) By Altering Source Files On Your Newly Forked GitHub Repository
This how-to guide provides an excellent description of this process: https://bitcointalk.org/index.php?topic=225690.0
Also contained in the linked guide are steps to creating your genesis block, and how to avoid common compilation errors.
STEP 5) Download Your Edited Source Code From GitHub and Follow These Directions To Compile Your New Wallet (Some Troubleshooting May Be Necessary)
STEP 6) Run At Least 2 Instances Of Your Software To Act As Nodes
If you’ve successfully made it this far and your block index is growing – then congratulations, you now have your very own cryptocurrency!