It will double the focal length of the lens from 55-300 to 110-600. You will also lose two full stops of light through the lens changing it from an f/4-5.6 lens into an f/6.7 to f/9.5 lens. In addition you will lose some amount of image quality, how much depends on how optically good the teleconverter is. On most cameras you will also lose the ability to autofocus since most cameras don't autofocus well at apertures above f/5.6.
In my personal opinion, unless you get a very good quality 2x teleconverter you are better off without one. I have a bad one that I wouldn't use for a paperweight. I know some guys using the excellent Nikon TC-IIE and it works very well, but not with all lenses.
If you drop your requirements down some there are other alternatives. A 1.4x teleconverter gives you 40% more length (77mm to 420mm), you will only lose 1.4 stops of light (which will allow your autofocus to still work in most conditions), and the loss in image quality is less. I use a Kenko Pro series 1.4x TC frequently and it works very well (notice that it is the PRO series and not the "Standard" series which, in my opinion, are terrible).