Austin - the image you're seeing is exactly what the 700D (or t5i here in the States) produces. Nothing is abnormal or wrong with your camera.
Some things to keep in mind:
(1) Higher ISO's will create more noise. I believe the base (i.e native) ISO for most Canon DSLR's is 160. Staying in multiples of that usually results in the "best possible picture." But again "best possible picture" does not mean no noise. It just means it's minimized.
(2) Kit lenses are slow are usually poor quality. Shooting at f/4 or f/5.6 means you need to up your ISO if you want to have a shot at exposing correctly with only minimal lighting (like you have in your video above).
(3) You need more light. All cameras need to be fed a steady and constant diet of light. The notion that people can buy a DSLR and run around in the moonlight (or on a cloudy day inside their room) and shoot images that are spectacular with no extra effort is a misguided notion perpetuated by too many videos from 2008-today. To expose well you need light. To run at a lower ISO, you need light.
(4) My experience with the ML hack is a double edged sword. It allows you the ultimate flexibility over the image in post, but it also seems to allow you to see all the flaws of these cameras and sensors. So running RAW means you're gonna see the good, the bad, and the ugly of a consumer camera. If you're not planning on doing some work in post with it (denoise, sharpen, color grading, etc) then it's probably not the right tool for the job.
Just my $.02
Ryan Holmes
http://www.ryanholmes.me
@CutColorPost