First off, anything 720p or above is HD, so 1600x900 is as well.
With ny capture program you want to record at or smaller than the resolution you play at, and even if they could render in a larger res, it would just destroy image quality. You have to start with enough pixels to fill the size of capture you're rendering, or things get really blurry.
So, if you want to have the rendered file show up as 1080p, you have to play at 1080p or more. Recording in 1080p is quite overrated though. The reason being YouTube made viewing videos so convenient that no one wants to take the time to download videos anymore.
It wouldn't be so bad if YouTube's viewer buffered 1080p video streams quickly, but it doesn't, which is why most don't use the 1080p option to watch YouTube videos even when it's available.
If you want to use YouTube, the best you can do with your current monitor is play at 1280x720 and set Fraps to record full frame. With DxTory however, you can play in one res, and capture in another. For instance you could play in 1600x900 and have it output the file as 1280x720, which is by far the most uploaded and watched HD size on YouTube.
DxTory is far superior to Fraps and only costs about $5 more.http://www.youtube.com/watch?v=W0KqsTa_HpM