Ok here goes:
You have a computer and a modem. To be able to install a wireless router you will have to do the following:
Go to where the modem is and look at the back of it. You should see (in most cases, unless it's a phone/data modem) three cables going in the modem: A coaxial (tv) cable OR a phone cable (depending if its a cable or dsl modem), a power cable (either regular power cable or a thin one coming from a power supply), and a network or usb cable that connects to your computer.
If you purchase a wireless router, make sure it is wireless BUT it supports wired connections (look on the back if it and see how many "holes" for network it has, if it's only 1, that's a strictly wireless router).
A good example would be a WR850G router from MOTOROLA.
Take a look at this picture:

Here are the ports on this router (should be very similar to yours):
1 is POWER.
2 is Reset button (this might or might not exist on your router)
3 is the 4 ports where your computers can be connected, also called LAN (local area connection) ports.
4 is the WAN (wide area network), in other words, the port where your modem gets connected
5 is the wireless Antenna
OK, first of, there is no such thing as a splitter for cat5 cable. That will not work.
Second, if you wish to install the router in the entertainment room, you will have to run a cable from the modem to the router, and connect your PC that is currently connected to the modem to the ROUTER instead.
Your connection should be something like this:
Modem -> Router -> Wired Computer(s) and/or other wireless devices.
How do you do this?
Follow these steps:
1. Run a cat5/5e/6 cable from the modem to the entertainment room. Connect this cable between the modem and your router (the cable from modem to router MUST be inserted into the WAN port - sometimes labeled as internet, extranet, etc. - on your router).
2. Run a cat5/5e/6 cable from the entertainment room to your PC that is now connected directly to the modem. Connect this cable in any of the other LAN ports (NOT the wan, thats where the modem gets connected) on the back of the router.
3. Connect any additional devices to the router if you have any more spare LAN ports.
4. Access the router configuration page and set up the router (this varies on modem, but if you require any additional help after you install this, let me know).
5. Connect any additional WIRELESS devices you have (given that you already set up the router, with or without authentication for the other devices to connect).
Good luck. Let me know if you need more help!