If you wish to use your own router and create a separate network from your landlord's, you should use a wireless bridge. Using the bridging feature in Windows as Jason Gordon has suggested would put your computer inside of your landlord's network and outside whatever network you created for yourself.
With a wireless bridge connecting to your landlord's wireless, your router connected to the wireless bridge, and your PC and other devices connected to your router you'd have a separate network and Internet access (and as an added bonus, access to all of the devices on your landlord's network, which they might not realize).
However, your router isn't directly connected to the Internet, so port forwarding won't work. The only way to expose a device on your network to the public Internet would be to configure your landlord's router to forward, say port 80 to your router's Internet-facing (public) IP address, then in turn configure your router to forward port 80 again to whatever end-device that traffic should go to. Additionally, their ADSL modem needs to be in bridge mode, otherwise it will be performing NAT on the traffic it passes to their router, defeating even the port-forwarding configured there.
All of this may not be possible for you unless your landlord is willing to let you make those changes, and even then, I'd be prepared for headaches and problems getting this to work since you'll have two routers forwarding ports...a non-standard configuration indeed.
Welcome to superuser. What sort of connection exactly is your landlord planning to make available to you? What sort of Internet connection is this (ADSL, Cable, Metropolitan Wireless, etc.)? – I say Reinstate Monica – 2014-08-07T02:26:31.627
They gave me the SSID and key to their wireless internet. They have a residential DSL internet service plan. – Glenak1911 – 2014-08-07T02:32:11.973
Can you edit your question to provide more detail about what you mean by 'I host my own site'? Do you intend to host a website from your network that needs to be accessible from the Internet, or do you mean something else? – I say Reinstate Monica – 2014-08-07T03:18:12.183
@Twisty I've edited the question. Please let me know if any more information is needed – Glenak1911 – 2014-08-07T03:24:10.627