First of all, you don't have to play a chord for every melody notes. Get you a nice accompanying chord progression to go with your melody and it should work out better for you.
Besides, for this song you could use simple 1-5, 1-4, and 1-4-5 progressions to fill it. Example:
1 happy birthday to
5 you, happy birthday to
1 you, happy birthday God
4 bless you, happy birth-
5 day to
1 you...
(repeat for "we love you we do")