Creating network off of exsisting WiFi

I am moving into a condo where they offer free WiFi, however there are two problems. One is the signal isn't very good in every room (anywhere from 2-4 bars), and the second is that I don't want all of my devices connected to the public WiFi.

My idea was to create my own network, which would be both wired and wireless, and then somehow connect my network to the open WiFi for internet access. I read that some wireless AP's and bridges allow you to connect a wired device to a wireless network, but I'm not sure if that would work in my case.

Here is a sketch of what the setup would look like:



Any thoughts?