Replacing Drive, with SSD (dont want duplicate info backed up)

I am replacing my drive for operating system with s SSD. The original drive is already backed up.
What steps should I take as I want to avoid having 2 copies of my system in the back up.

by 2 copies I mean I do not want to end up with the original copy of my regular hard drive and then another copy of my Solid State Drive.

Any suggestions on how to go about this process.
I had planned to replicate my original drive to the SSD, then format my original drive and set up directories for Data (ie downloads, Pictures, Documents and Music)

Has anyone done this?