I have an idea for you. This might work. You REALLY need to read up on Copyright Law, though. To be honest, you might not be the best person for this. You may want to speak with an experienced copyright person. Anyway...
This might work: Stop broadcasting the video. You make an audio only CD where you and friends play the parts in the movies. The customer is instructed on when to start the audio CD so that it syncs with the movie. You can also distribute a book that tells the person when to start and stop the movie/audio and then do your thing with your company.
So all you are selling is an audio CD, which you performed, and a companion workbook.
This MAY work because what you are doing is making a "cover song" of the movie audio. You will need a mechanical license, if it is even available. http://en.wikipedia.org/wiki/Mechanical_license
I have never heard of this for movies, but it might work. A mechanical license will be cheaper and easier than negotiating with a movie studio.
Rifftrax.com (some of the MST3K stars) has been putting out very good voiceover tracks for current movies. You might ask them what kind of licensing they needed to go through.
They probably don't need any license. What rifftrax is doing is selling you an mp3 with their own comedy on it. It is a completely new work. The customer buys/rents the companion DVD and then syncs.
That is slightly different than OP because OP wants to broadcast the movie, or at least the exact words. The exact words are what he needs to license, even if he does perform them himself.
If OP can figure out a way to teach people language by making his own rifftrax mp3 then he is in business. Good call prodigal_erik.
Rifftrax guys gave me an idea - what if I setup Amazon shop and will sell legal DVDs. People who buy DVD get access to the same movie on my website (invitation code). Will it work?
The reason why is at the very heart of copyright law. You need to learn this area.
Owning copyright to a work gives the owner EXCLUSIVE rights to distribution of that work.
Therefore, just because you sold a legal copy to a user via Amazon you do not have the right to broadcast the work on your website. Only the copyright owner can do that, because he has exclusive distribution rights. Your broadcast isn't authorized even though you selling DVDs is authorized.
Be careful with this stuff. In my opinion you are playing with fire here. The RIAA has no problems with burying people who distribute songs illegally. MPAA has done less of this, but you don't want to be the example.