This is the path I'm going down Transistor Simulator And Calculator
That is not quite how you want to use it.
R2 should be zero ohms, that is it should not be in the circuit, replace it with a short.
R1 is your load, this is where the relay coil should go.
You need a resistor from the base to your arduino PWM output.
This will explain what PWM actually does.
http://www.thebox.myzen.co.uk/Tutorial/PWM.html
This shows you how to wire up a relay:-
http://www.thebox.myzen.co.uk/Workshop/Motors_1.html
Any general purpose NPN transistor will do, I am assuming your relay doesn't take current, less that 100mA anyway. If it is more you need a transistor that can handle the current your relay requests.