What you want, is a Selenium Grid.
There are at least 2 separate entities that you need. You need...
A Grid server
A Node
In your case, what you will do, is for your RDP machine that you have access to, you'll want to run your selenium-server-standalone.jar
with the -role node -hub http://my.hub.ip
arguments attached.
For the hub (should be either your computer, or another dedicated server) is run the selenium-server-standalone.jar
with the -role hub
arguments attached.
Then what you'll do, is when you are on YOUR computer, write your test, you can fire tests at the grid, which will fire the tests, in turn, at any available node. In your case, your RDP.