You could play any number of different C6 chords: some using barres, some higher up the neck, chords with or without C in the bass and even chords that double an open string note. But practically, it makes sense to play the C6 shape that is closest to the common open C shape, as this is probably what you are most familiar with, and as it will be close to other chords you are probably familiar with. I have suggested three ways to finger this below (sorry, they were scribbled in a hurry!):
The first fingering is like an Am chord with the fourth finger adding a C root note. The second fingering is that suggested by Meaningful Username. The third fingering is one I quite like; the second finger bends "backwards" to barre two strings at fret two. This takes a bit of practice and finger strength, but does mean the rest of the chord is a regular C chord, making it easy to move C to C6 to C7, which happens during this song.
A couple of things to note...
This chord shape doesn't have the note G; the fifth of a C6. In theory, this makes it an Am/C chord, but in practice it doesn't affect the sound of a C6 dramatically.
The music you link to is just a lyric sheet with chord symbols - it doesn't necessarily give accurate information about the chord voicings (shapes) that would be best to use. Also, the music is not conventionally published, so may not be correct anyway! Having said that, if C6 is the correct chord to use in this song (sorry I haven't had a chance to listen to the original to check…), this C6 shape should work well, both to listen to and to play.