Try sudo apt-get install isc-dhcp-client, which will install the dhcp client, though it won't explain why it wasn't installed to begin with. This also presumes that your problem is a lack of said client.

If that doesn't work, you should look into using the tcpdump command on one of the interfaces and post back here with the results. For example, sudo tcpdump -i eth0 -n should show you if there is any traffic passing over the eth0 interface, though you may need to correct for the right interface.

Since tcpdump didn't see any traffic, start the tcpcommand again, unplug your LAN connection, wait a couple seconds for the computer to realize you've unplugged it, then plug it back in and look at the tcpdump output. There should be several DHCP request sent out by you're computer, and under ideal circumstances, responses to those queries as well. If you don't see anything at all, then perhaps DHCP is request an address on the wrong port, maybe check DHCP configuration. If after checking your DHCP configuration you're sure that DHCP is requesting an address on the correct port but that you're not seeing any traffic with tcpdump, check the logs for any errors related to DHCP or errors transmitting network traffic. If nothing is obvious, post the logs here.

Beyond that, I don't know what else to do to diagnose this, perhaps your suspicions of faulty hardware were correct, or more unlikely, driver trouble.