Firstly you don't need Bind or similar unless you are actually hosting domain names. If all you're after is a DNS resolver type of server then yes you need to use the Mikrotik as your primary DNS server.
Basic steps.
/ip dns set primary-dns=208.67.222.222 secondary-dns=208.67.222.220 allow-remote-requests=yes
These are the OpenDNS servers, you don't necessarily need your own ISP servers for DNS to work unless they specifically block/throttle 53 requests.
Then create this type of rule;
/ip firewall nat add chain=dstnat action=dst-nat to-addresses=192.168.1.1 to-ports=53 in-interface=lan dst-port=53 protocol=tcp
and
/ip firewall nat add chain=dstnat action=dst-nat to-addresses=192.168.1.1 to-ports=53 in-interface=lan dst-port=53 protocol=udp
Note that
represents the internal IP address of your router and that
represents your local ethernet interface on your router.
This should work. Let me know if you need more help.