You can install the RC with e.g.:

pip install "astropy[test]==5.3rc1"

and test it with:

python -c "import astropy; astropy.test(remote_data=True)"

This will produce a long sequence of dots and letters indicating the test is running. At the end, you should see something like

===========  23723 passed, 239 skipped, 124 xfailed, 1 xpassed, 2 pytest-warnings in 476.85 seconds ===========

If this shows no failures, you can report it in the table below as a "success", otherwise, put it as a "failure" (and if need be you can create an issue in the astropy issue tracker). You can also check that test suites for packages you maintain pass with this RC.