You mention both routers and access points in your question. An access point bridges wireless media to wired media (with many optional bells an whistles, too, on a real access point). A "wireless router" provides access point functionality along with additional functions to route TCP/IP between subnets (and provide network address translation, etc). I don't think you're looking for a second TCP/IP subnet and routing between it and your existing LAN subnet. I think you just want an access point, and you can use another router as a cheap access point.
Go get a second wireless router, and configure it as follows:
- Plug one of your computers into the new router's wired Ethernet ports w/ nothing else connected on the new router.
- Configure the new router to disable its DHCP server, and with an IP address that's in the existing TCP/IP subnet that your current network uses.
- Configure the wireless Ethernet functionality on the new router to match your existing router's SSID and encryption settings.
- Set the WAN interface on the new router to "DHCP client" and forget about it -- put a piece of tape over the port or something similar so you never plug anything into it.
- Disconnect your computer from the new router and plug one of the LAN ports on the new router into your LAN.
- The new router will bridge the wireless segment to the wired LAN, and clients of the new router will get DHCP and Internet access from wherever current wireless clients get DHCP.
We've done this fairly frequently with one Customer who wanted access points "on the cheap" by using Linksys WRT54G wireless routers and, through the procedure above, effectively disabling their "routing" functionality and just using them as wireless access points.
Alternatively, if you want to spend a little more money go get a wireless access point that doesn't have router functionality. It will need less configuration but will be a little more expensive. (Market dynamics are at play there... There's less demand for low-end access points than routers so the price of an access points ends up being higher than a router even though an access point "does less".)