Best way to track visitors to a website as an affiliate?

Hi there guys,

I wonder if anybody could help me out with the following.

I am going to be promoting a recruitment company (recruiting doctors basically) and my job would be sending leads to the site.

Now, I'll get paid every time I get a doctor registered with the company and I am trying to work out the best way for the company to know that "I" sent those leads (so I get paid right? )

I thought about creating the same website with a different domain name so the leads gets to "me" when they register - and then - I can email my client the new recruit's details to get registered in their books. So in a sense I would not promote their website but another one identical that I would host myself with a different domain name.

Last think I want to do is to create a campaign, specially through social media, where the prospects visits the website directly and they sign up without knowing where they come from.

The other alternative would be to ask them to create an affiliate link for me and I can then redirect from my own domain but I doubt they know how to do that. If that was an option how can I ask them to create the link?

Any ideas that I may miss?

Many thanks,

Alvaro