Thanks for using Apple Support Communities. The Remote app does not appear in the iTunes library; instead, the iTunes library appears in the Remote app which you can then use to manipulate that library. This article has some information that might help:
This page also has some additional information:
I had this problem for weeks as well. The way I finally managed to do it the other day was simply by opening the remote app on my phone first and then clicking on 'add an iTunes library' within settings which then gives you a code. Only then do you open iTunes on your PC and then it magically appears next to the iTunes buitton in the top right hand corner, like you said. Then just click on it and put in your code and you should be connected. (I am assuming of course that you have home sharing switched on on both your phone and iTunes on your PC)
Previously when I tried it I had already had iTunes open on my computer and the remote button just didn't appear. I would be interested if this works for everyone or whether I was just lucky!