There is a vi hart video on youtube about happy birthday. She plays simple chords in this progression: C, G7, C, G7.

But when I think of happy birthday I think of something much more complex. For the melody I think of this:

C, C, D, C, F, E, , C, C, D, C, G, F, , C, C, C, A, F, E, D, Bb, A, F, G, F, and then same thing an octave above

Note: I use 2 commas with no note in between to designate a pause

And for the accompaniment I think of Chords played like this:

Root first, then 3rd and 5th at the same time

and the chords going well with the melody.