Say you want to use DNS servers 184.108.40.206 and 220.127.116.11.
In /etc/resolve.conf, have the following line:
nameserver 18.104.22.168 nameserver 22.214.171.124
and make sure there are no
dns-nameservers lines in /etc/network/interfaces.
In /etc/dhcp/dhclient.conf, have the following line:
supersede domain-name-servers 126.96.36.199, 188.8.131.52;
Restart the networking service:
$ sudo /etc/init.d/networking restart
Check if the DNS server is used:
$ dig somehost.com ... ;; SERVER: (should see the DNS server here) ...