Not sure about the original key but this is how I play it:
[Verses]
C Am7 Dm7 G
I'm singin' in the rain
Just singin' in the rain
What a glorious feelin'
Dm G7
I'm happy again
Dm7 G7 Dm7 G7
I'm laughin' at clouds, so dark up above
Dm7 G7 C G
The sun's in my heart and I'm ready for love
[Verse]
C Am7 Dm7 G
Let the stormy clouds chase
Everyone from the place
Come on with the rain
Dm G7
I've a smile on my face
Dm7 G7 Dm7 G7
I walk down the lane with a happy refrain
Dm7 G7 C
Just dancinin' and singin' in the rain
Source: https://tabs.ultimate-guitar.com/tab/gene_kelly/singin_in_the_rain_chords_1131961