I want to get into taking more bird images, and am on a very limited budget. I am wondering if I can use the Canon 2x Extender on my Rebel T4i camera's EF-S 18-135 kit lens. If so what sort of price range am I looking at?
No it is not going to work with the EF-S 18-135mm.
I don't know of any EF-S lenses that the 2x extender works with. The lenses are too close to the body and would interfere with the area that the extender takes up. More info on why only certain lenses work with the extenders can be found here: Why can only certain Canon lenses be fitted with an extender?
Here are compatibility charts for what does work:
Since some of the optics of the Canon 1.4X and 2X extenders extend into the back of the lens attached to them, they are only compatible with specific lenses. No EF-S lenses are compatible with any version of the Canon Extenders.
Here is a link to the list of Canon lenses compatible with Canon extenders.
Note that the only lenses with focal lengths below 135mm that are compatible are the 70-200mm L series variants and the 100-400mm L. All of the compatible prime lenses are also L series lenses. One of the downsides of teleconverters is that they magnify the flaws of the lenses with which they are coupled. If you don't start with a fairly high quality lens to begin with the final image quality can be rather disappointing.
You do have other options, though. Your most budget friendly option would be to buy a longer telephoto lens rather than try to use a teleconverter with your 18-135mm. For about the same price as the Canon EF 2X III you can buy an EF 70-300mm f/4-5.6 IS. You would get the benefit of better optical quality and the f/4 to f/5.6 aperture range. If that is still outside your budget you might consider the EF-S 55-250mm f/4-5.6 IS II. Note that any EF-S lens would not be able to utilize any teleconverter of which I am aware.
Although I wouldn't recommend using a 2X extender with the EF 70-300mm f/4-5.6 IS, it could probably handle a 1.4X teleconverter from a third party such as Kenko fairly well. You would lose one stop in terms of aperture so the f/4 end would be effectively f/5.6 (still usable in terms of AF) and the f/5.6 end would become f/8 (probably not usable in terms of AF).
This Kenko 2X teleconverter (also marketed under several other brand names) will attach to any EOS camera and allow any EF lens to be attached to it. Since you take a two stop hit in terms of aperture when using a 2X teleconverter, auto focus will not work with lenses slower than f/2.8 or possibly f/4 depending on the camera model. f/2.8 lenses effectively become f/5.6 lenses when coupled with a 2X extender, f/4 lenses become f/8, and so on. I've been successful using AF on Full Frame Canon bodies with a 2X Kenko and f/4 lenses, but not so successful when trying to use the same teleconverter and lenses on an APS-C body.