I have replaced the sink strainer on the kitchen sink and used plumbers putty. But the putty keeps oozing out whenever there is hot water in the sink, which causes the drain to loosen and leak. What am I doing wrong?

You did not tighten the large nut under the sink enough. When installing a new drain you neeed a roll of putty no thicker than a pencil, wrap the drain and install the washer and tighten until the drain is almost totally seated againist the sink. Wipe away that putty and that should be the end of the issue.