We just had the exact same request, we bought a little app called Captionate (PC – http://www.captionate.com) that is helping us out.
We had to export a flash file (flv) of our movie and combine that with the xml file provided for us by our Closed Caption house. We then were able to marry the two with Captionate (PC only, unfortunately). The program was very intuitive and easy to import/export. Costs $60.
Here’s the issue we are just now getting so. Through captionate you can export a .flv, data .flv or data .xml. I just sent them over to the webmaster to see if he can use them as is. If that doesn’t work we will need to provide him with a “skin” or an embedded player that will allow the user to toggle the Closed Captions on an off.
If you DO need to export with a skin, the only way I am aware of so far to do that is through Adobe Flash (V. 8 or above). The process sounds simple enough to do without knowing Action script. You can find FREE skins and a tutorial on how to do that here:
https://www.adobe.com/devnet/flash/articles/video_captionate.html
That page also has a short video on captionate and so on. Hope that helps!
Have a wonderful day.