To answer your question:

I'm not sure how to do this but is it possible to get the identity of the computer(like the computer's name) the first time it's run and then the next time it's run the program could look for that identification or else it won't run


You can easily do that by retrieving any relevant information like the computer's name and then store it in the registry, encrypted file or anyhow. But I guess that would be done on the first run or first install - which won't avoid copying your application and using it on another computer.