First, follow all instructions on for setting up a network of WP sites. These instructions can be found here: WordPress Codex: create a nework.
Then, install the WP MU Domain Mapping plugin.
Step 1: add a new site
Now you’re ready to add a site and map it to your domain. Go to My Sites > Network Admin > Dashboard > Sites and Click on Add New. Here you’ll have to write something in the “Site Address” field – this will be “reverted” in Step 3.
Step 2: domain mapping
Go to My Sites > Network Admin > Dashboard > Settings > Domains. If your new site is your first “sub-site” in the network, ID will be 2, and so on.
Step 3: real domain name instead of subdomain
Now go back to My Sites > Network Admin > Dashboard > Sites and edit your site. You can now change the “Domain” field to match the real domain name of your site.
Here are the two most useful article referring to adding new domains to WP Network and mapping them to domain names:
How to map domains to sites within WordPress multisite
WordPress Multisite Beyond Basics: Essentials and Domain Mapping