The steps are simple:
Broadcast using the suitable mask on the local network.
All the local machines will respond with their IP address (it is possible to configure your PC to ignore broadcast queries)
Using the list of IP address, ask all the machines, one at a time, their ID (computer name, mac address etc)- it is again possible to fake your ID or ignore the query.
Select your machine from the answers received - it is not difficult.
When the server sends a response to the DHCP request, the block will be available in some file (are you using windows or linux?)
You can just scan the file and figure out which machine is which...