As you'll have doubtless noticed, when it comes to the best monitor with webcam your options are somewhat limited. However it comes down to two basic criteria: the quality of the monitor itself and of the webcam.
With the webcam, ideally you want something that delivers the full 1080P. While 720P is just about acceptable, that extra helping of resolution makes a real difference (except when you're reduced to a tiny portion of a gallery view). Stereo microphones are a definite plus, and if you value your privacy then you'll want either a pop-up webcam that you can put away when not in use, or at least one with a privacy screen that you can put down over the lens.
As for the monitor, for now you'll likely have to compromise. There are very few 4K options available so you're pretty much limited to 1080P HD; to make sure you're getting the best quality, though, look for a decent refresh rate so that everything's nice and smooth, a fast response time for the same reason, a wide viewing angle and good brightness and contrast. You'll want connectivity that goes with your computer – VGA, DVI, DisplayPort or HDMI – and while built-in speakers can be a bonus, they rarely kick out much power so you'll be better off with desktop speakers or headphones.