If you do not want CS3 on the computer delete it BEFORE you install CS6. Otherwise you can run into file association problems when deleting old versions.
When you install CS6 it will ask for the qualifying product. Type in the s/n of CS3.
There is no conflict with having the 2 versions on the computer, in fact most users perfer this as they have a backup if one has a problem.
Be advised that CS6 takes a lot more horsepower and puts more demand on computer than CS3. It really needs 64 bit OS and min. 8 gigs of RAM to work well.