No, that's not what you do. You will need to get a router to do what you want. It might be worth looking for a router that advertises compatibility with the PS3. I can't honestly say how much of that is just a marketing scam, and how much of it is actually useful. Anyway, you connect the router to the cable modem via the ethernet jack, and then your computer and PS3 connect to the router also via ethernet.
The router will likely come with a short ethernet cable to let it connect to the cable modem, but you will need to purchase some standard (patch) cable for your PS3 and computer, and a PCI ethernet card for your computer if it doesn't already have an ethernet controller.
I use a Motorola Surfboard Cable Modem to connect my computer to the Internet. I currently have a splitter on the white tv cable coming out of my wall - one lead goes to the tv and the other goes to the modem. The modem is then attached to my computer via a USB cable.
However, I noticed that my Motorola Surfboard Cable Modem also has an Ethernet cable connection on the back of the device.
Do I simply hook my PS3, using an Ethernet cable, into the cable modem to be able to go online with my PS3? If not, what do I need?