DHCP is what issues dynamic IP addresses to computers and other devices on the network.
DNS is what computers and other devices on the network use to find each others IP address. DNS is also used to find the domain controllers as well as other key services on the network.
At least one DNS server must exist on the network in order to make active directory work. At the other hand, DHCP is not really necesssary.
At first sight, one could say that DHCP has no direct relation to DNS, but DHCP can also be configured to dynamically collaborate with DNS to provide secure dynamic updates for A and PTR records in DNS.