This part isn't always so easy! YouTube has lots and lots of Russian movies with subtitles in English, but far fewer with Russian captions for the deaf. However, if it's a relatively short clip, there are people on here who are willing to type up a transcript. Or, in some cases, the Russian script/screenplay for an entire movie may be online (if it's a very popular movie), and people here may be able to help you locate it with Google.
Fortunately, it's not so difficult to find videos of Russian songs with "karaoke lyrics" that you can read as you listen.