Background verification concerns

No, you should not resign until you have a firm offer with a start date that is not conditional on anything, not even background checks.

You can of course sign an offer (indicating that you will accept it if confirmed) before then. You then resign once the offer is unconditional.

The biggest issue is if the offer has a start date on it that may not be achievable if the background check is not completed in a timely fashion. Do not sign an offer that has a fixed start date unless the company guarantees (in writing) that the background check will be finished in time to let you serve an appropriate notice period with your current employer. Instead ask for an offer that gives a start date something like "two full weeks after I am informed of the successful completion of all background and suitability checks", for a two week notice period.

Also ask what will happen if the background check is not complete by the start date. Will they allow you to start anyway?

Since this is India you should also ask what will happen if your former company does not give you a relieving letter because you did not give them enough notice.

If the new company objects, ask if they would be OK with being treated like that (i.e. being given no notice) by someone leaving them.

1. Never quit a job until everything in the new job is ready to go

2. Background checks are not about how you did in the last job. In there US, a background check merely verifies your employment at previous companies. The background check is to look for criminal activity or lies in your application (did you really get that degree). No one cares that you didn't finish work when you left a former company. You are at risk because no one can predict how a background check will go, but how you left your former employer isn't something you need to worry about.