Perhaps I can clarify some of the other answers.
1) Setup the second "router" as an access point, or repeater. In most hardware, you don't have to disable DHCP, it will be disabled when the box is in access point mode.
2) Since this box is a repeater, if the first router is providing DHCP for the wired ports (which it should be), then computers on the wireless repeater will get DHCP from the first router. If you have any doubt, plug a laptop into the wired port on the first router and verify that it gets an address on 192.168.1.x. Make sure your router has enough IP addresses setup in DHCP config to assign to all your new computers on the repeater wireless.
3) Make sure you setup the wireless channels to be separated as much as possible, while still providing good coverage and avoiding interference with neighbor's wireless.
4) Configure a static 192.168.1.x address on the second "router" so you can always access its web interface on a known address and can put a link shortcut in your browser. You might need to configure the DHCP on the first router to accomplish this, but some DHCP servers will honor any static IP that they encounter.
5) If you haven't already bought the second router, try re-positioning the first router to get broader coverage. Maybe you don't need to buy the second router. Remember that altitude is key, as well as avoiding metal walls and aquariums. Also try switching channels for better coverage and less interference. (Sorry if this is too elementary.)