Most of my TT boxes are connected this way to the broadband at the location. The Verizon wireless kit TT offers is sweet, but they will biff you another $20 or so per month on top of your fees.
How are you connecting the TT PC to the router? Cat 5 or are you trying to go wireless to the router/WAP? I've had limited sucess using a wireless bridge on a TT box, it can be a major PITA to get it working and the few I've done I have had to go back a reset a few times.
Straight cat 5 run to the router or switch is the best way to go. If you have a cat 5 run going, and it's connecting to the network, but won't connect to the TT server, check the broadband status information see what gateway and DNS numbers got passed to the box. One location I had used a mess of a network on top of another network (multiple networks/subnets) for point of sale and credit card system, and I had to forget using DHCP and manually setup Ip address, DNS and gateway. Although you may be connected to a "network", if the gateway isn't configured properly the box will never find it's way out to the internet. Some places I've setup were a complete screw in terms of the network, sometimes you have to sort out the network first. A laptop would be handy and help in determining the network layout.
Also, the box needs ports 80, 443 and 5223 opened, so if there's a firewall in place anywhere, make sure those ports aren't closed.
D