Can you say for sure that the 192.168.0.* subnet is the correct IP range for both routers? Here's how I think it should be set up logically. Note, you should disconnect all computers from either router whilst configuring, to avoid IP address conflicts:

Router 1 (ADSL router/modem, connects to the internet, I'm assuming):
- IP address: lowest IP on the appropriate subnet. This will be 192.168.0.1 or 192.168.1.1 most likely. I'll assume 192.168.0.1
- Subnet mask: 255.255.255.0. This defines the subnet as covering all IPs in the range 192.168.0.*
- I recommend turning DHCP on in this router.

Router 2:
- IP address: This should be any IP that's addressable by router 1, that is, any IP in the range 192.168.0.* where * is not 1 or 255. Let's use 2.
- So your router 2 has the IP address 192.168.0.2, we want to keep the 192.168.0 the same always, so use the subnet mask 255.255.255.0.
- You should have DHCP turned off in this one. The idea is not to have this one actually doing routing.

Xbox:
- Similarly, you just need to make sure the Xbox is on the 192.168.1.* subnet, where * isn't 1, 2 or 255. Let's make it 3.
- Subnet mask should be the same. There's no real reason you'd ever need a different one than is the default for your router.
- Gateway: I believe you should use the IP address of router 2 as your gateway here.
- DNS: You have a couple of options. You could find the DNS server addresses for your ISP. You could use OpenDNS addresses. Or you could use 192.168.0.1 and allow router 1 to handle DNS.

To test, you could take the Xbox network cable and plug it into a laptop or desktop computer and see if the computer is able to access the interweb.

Brief explanation of subnets:
They're used to divide networks into smaller, more manageable subnetworks. Typically, a device on one subnet can only see devices on the same subnet. Here's an example of subnetting:

Ok, your router uses the IP address 192.168.0.1/24. This means that the first 24 bits of the IP address (8 bits x 3 bytes, which equates to the first three numbers; 192.168.0) will be the same for every device on that subnet. You could then define two subnets, 192.168.0.1/24 and 192.168.1.1/24. Which means that you'd then have two subnets covering the IP ranges 192.168.0.* and 192.168.1.*.