What is the process of marriage registration after court marriage

You can register your marriage under Hindu Marriage Act. Here are the procedures:

For Marriage Registration under Hindu Act: You can apply at office of the Sub-Divisional Magistrate in whose jurisdiction the husband or wife resides on any working day.
Fill the Application form duly signed by both husband and wife.
Verification of all the documents is carried out on the date of application and a day is fixed for the appointment and communicated to the parties for registration.
On the said day, both parties, alongwith a Gazetted Officer who attended their marriage, need to be present before the ADM. The Certificate is issued on the same day.

Dear Sir, it depends upon state to state basis, where you want to get the registration done. Since both of you are engaged already , it would be advisable to have a personal discussion with you, then only legal opinion can be provided.

No need for marriage registration after court marriage. Marriage registration is only required when marriage is done by following any other ritual in different religions but when you are doing marriage through court than separate registration is not required.