I am wanting to use my Server 2012 machine for DHCP. Once I add the DHCP role, what should my router be configured for? I have a cable modem bringing the internet in.
Would the server replace my router?
Your existing router doesn't have to be the DHCP server for your network while still acting as a router for your Internet connection. Those two functions can be separated.
When you configure the Windows DHCP role you'll need to insure that your DHCP scope includes the "Router" option with your router's IP address specified. This will allow DHCP clients to receive the router's IP as their default gateway.
You'll want to disable the DHCP server on your router so that it doesn't also attempt issue leases.
If you're using Active Directory on the Windows Server 2012 machine you'll want to configure the DHCP scope to provide the server's IP address to clients for their DNS server. If you're not using Active Directory then you can still run an DNS server in-house if you prefer, or you can direct your clients to a public DNS server at your ISP or other public host.