I think the answer you're looking for is here.
Blatantly plariarised:
You can add the following line to /etc/dhcp/dhclient.conf:
prepend domain-name-servers <working DNS IP(s) here>;
This adds the DNS IP address(es) you specify before that/those provided by the DHCP. If you would like to add it/them after the address(es) provided by the DHCP, just use
append domain-name-servers <working DNS IP(s) here>;
If, instead you would like to ignore the DNS address(es) provided by the DHCP altogether, use
supersede domain-name-servers <working DNS IP(s) here>;