Verify that your router will let this computer get its config via DHCP. It sounds like it might be getting refused.
Here's what mine looks like:
# Generated by NetworkManager
My router is 192.168.0.1
It forwards the DNS requests to the DNS servers it has set up within its config.
Another good test to verify connectivity to the outside network is to find the IP address of a computer and try pinging it by IP.
root@pirate-chest: ping 126.96.36.199
PING 188.8.131.52 (184.108.40.206) 56(84) bytes of data.
64 bytes from 220.127.116.11: icmp_seq=1 ttl=242 time=50.7 ms
64 bytes from 18.104.22.168: icmp_seq=2 ttl=242 time=46.4 ms
64 bytes from 22.214.171.124: icmp_seq=3 ttl=242 time=44.5 ms
64 bytes from 126.96.36.199: icmp_seq=4 ttl=242 time=46.0 ms
64 bytes from 188.8.131.52: icmp_seq=5 ttl=242 time=46.2 ms
--- 184.108.40.206 ping statistics ---
5 packets transmitted, 5 received, 0% packet loss, time 4017ms
rtt min/avg/max/mdev = 44.579/46.819/50.770/2.087 ms
Obviously, you'll need to do the nslookup on a computer with a valid connection to the internet. Or, you can use the IP for slashdot I've provided above. If that test fails, then you know that not only is DNS broken for your system, your router is refusing traffic from this machine to the outside world.