You have two main options (aside from using a 3rd party DNS host)

1.) Set up branded nameserver names and use the DigitalOcean DNS tools to manage your domains. - This tutorial will walk you through the process of pointing your nameservers so you can use ns1/2/3.digitalocean.com with your own (ns1/2/3.yourdomain.com) domain name.

2.) If you prefer to host your DNS services within your droplet you will want to:
a.) Install a DNS server on your droplet. BIND is the industry standard and most popular option
b.) configure your nameserver registrations (with your registrar) to point to your droplet’s IP
c.) set up zone records for each domain you will host