Do I need to have a home network?
You don't need a home network to play on Xbox LIVE. You can connect your Xbox console right to your cable or DSL modem with an Ethernet cable. That's a direct connection. It's fast and simple to set up. But there are disadvantages to the direct connection:

Your PC cannot access the Internet if your Xbox console is using the connection.
Unplugging the Ethernet cable from your Xbox console and back into the PC every time you need to use your PC gets old real fast.
Unless your TV and Xbox are right next to your PC, you'll have lengths of cables underfoot. The second time you trip over them, you'll want to find another solution.
Note
In most cases, you will need to use a PC to initially configure your high-speed Internet connection before directly connecting your Xbox. Contact your Internet provider for details.