Those kind of videos are being done quite often these days.
In general, to add people or other items to a scene like that, it will require a couple techniques.
1. May require shooting people on green screen to overlay them on the footage unless it is a CGI item.
2. Unless positioned in a place where nothing overlaps it, there will be some roto work required (example, in order to place behind a person of object existing in the original scene.
I try to provide examples when I have them or make something and just so happens I have one for this too. In the link below, I shot myself on green screen, integrated myself in to the scene. Now, around my waist and hands, I had to mask/roto myself out so that my body did not show in front of the bottle on the table, etc. (I'm the guy with the brown coat next to Doc).
Hope this helps.
There are several operations involved. Basically you have to think in layers. Everything your inserted actor passes behind is a layer, everything your inserted actor is in front of is a layer.
In your pre-production planning do you have to consider lighting, shadows, camera angle and distance from this subject. All of these must match to pull off a seamless composite. If there are camera moves then you have to both understand how to track camera movement or background movement and how to shoot so you can get accurate tracking information.
There are several techniques for separating items from the background and these include rotoscope, track mattes, procedural mattes, and keying.
If you are introducing CG elements then you also have to learn how to capture the lighting in your master scene so that you can re-create it with your 3-D models.
That sounds like a lot to know and learn, and it is.
To get started I would suggest that you plan your shot using a locked off camera, measure the distance from the camera to the critical elements in the scene, then shoot a "clean plate" without any actors, then shoot your master with some actors, then shoot another scene with just The actor you want to add to the scene.
If you want to you could shoot your second actor using green screen to make it easier to separate them from the background. Get something simple like this to work and you're on your way to learning how to do the more complicated shots.
Make sure you have a very good understanding of the basics of After Effects, compositing, and camera work.
One last thing. If I were doing "Fifty Shades of Tubby" every single shot would be a separate composition. There is a real tendency by newbies to try to use AE to edit. You should only be creating single shots or on rare occasions short sequences in AE. Your editing should be accomplished in a nonlinear editor like Premiere Pro.
I'd just add to Rick's great post by also saying, the quality also depends a lot on what you are going for. Is it serious, is it a parody? There are tons of funny mash ups out there that basically just slap someone's head on another with some tracking and it's funny in part because it's also dorky.
If you are going for something a bit more serious/realistic like the example I posted where I was actually trying to blend myself in to the scene, then there's much more to do.
Also agree with the comment about not using AE as an editor. For very short sequences, I often do some editing there but first it's put together in Premiere then select, copy, paste in to AE. Very good work flow for small sequences.
What I'm most interested in is how did they extract the teletubbie from the teletubbies episode and insert it into the 50 shades of grey trailer? I will be doing similar types of mashup videos combining two different pre-existing sources.
Oh, ok, that's more clear. So, I believe we've answered the second part of your question. Now to the first part, how they got it out?
Assuming they pulled it from an actual Teletubby clip, there's really only one way. Rotoscope.
With the fairly simple shape and good contrast, small sequences would not be terribly difficult.