I personally have a router between my ADSL modem and my PC. The router has it's uplink port set to use whatever IP address the modem requires (can't remember right now how exactly that's configured), and provides DHCP to all it's downlink ports. My computer is set to use DHCP. The rest mostly takes care of itself.
If your computer is plugged directly into the ADSL modem, then it's likely that your PC is set to use a static IP (on the same subnet as the modem). In Windows, look at your settings by going to Start->Run and typing "cmd". At the command prompt, type "ipconfig /all".
Look at the settings for "Ethernet adapter Local Area Connection". Determine if "Dhcp Enabled" is Yes or No. If it is yes, then TinyCore probably would have worked right out of the box, so it's probably No.
If it's No, then you have a static IP address. Write down the IP Address and Subnet Mask values. I'm not sure if you'll need the Default Gateway, but if that's set, write it down too.
Now start TinyCore Linux up, and open a terminal.
You'll probably have to kill udhcpc, so type "ps" to find it's PID, then type "kill <pid>", where <pid> is udhcpc's PID.
Type "ifconfig eth0 add <address>; ifconfig eth0 netmask <mask>;" where <address> and <mask> are the values you previously wrote down.
After that, everything should work. Of course, all that assumed you had a wired connection to the modem.