In this song we are going to learn how to pick out melodies and associate chords with the melodies. Let's start the song on a G. Sing the song in your head and the last note of the song is where your ear kind of pushes you toward G. In the beginning it's hard to figure out what that note is, so let's just start on the G and see where it takes us.
If we start at the G and pick out the notes of the melody until we get to the end, we'll see that the song ends on a C. We are going to assume, based on that, that C is the key the song is in. Let's start at the beginning of the melody, playing a C chord in our left hand. It seems to work, but the chord wants to change when we get to the second "falling down". We hit a D in the melody line at that point. The most popular chords are root, fourth, and fifth. We know the root doesn't work. Let's look at the fourth and the fifth chords and see if we can find a D contained in one of them. Sure enough, the five chord does. So let's switch to a G at that point in the song. When the melody gets to the E notes, let's switch back the C chord, since that one contains an E. We'll go back to G when the melody gets to the "my fair lady" part.
Now, what we have learned about the song is that it is in the key of C, it starts on the fifth interval of the C major scale, and it used the root and fifth chords. Armed with that knowledge we can play the song in any key we want. Just find the fifth note of that scale, and play the same pattern within that scale instead of the C major scale.