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I think providing a short answer and a long answer might be appropriate here. I'll do the short answer first :)
Short Answer: Literature databases like Literary Reference Center or Gale Literary Sources (both linked below) would be very effective for finding more authoritative criticism about Shakespeare's Hamlet. Because it's focused on literary criticism, you can search for criticism about a certain work (Hamlet) or about a certain author (Shakespeare) or just by keyword (looking for your term anywhere).
Long Answer: Library databases are collections of organized information, sometimes about very specific topics (like literary criticism, biology, or careers, for example) and other times about multiple topics (we call these multi-subject). Whether these databases are specific or broad, they will contain articles from journals that are high-quality sources of information, authored by very knowledgeable, credible people. Searching through a database can let us find that information easily.
Before we can begin searching through a database, though, we need to choose an appropriate one. Because they are sometimes specific in scope, not every database is equally useful for every assignment. This is where a friendly librarian can be very helpful! But if you're unable to speak to a librarian, the library organizes our databases in a few different ways -- by database name, by database subject, and by database type. If you know exactly what database to use, look by database name; if you know what subject you're looking for, choose a database that falls under that subject (choose "English" from the "Subject" drop-down list at the top); or if you're looking for background information, you might want to look at database type (like encyclopedia) to get you started.
To access any HCC library database from off-campus, you'll be asked for your HCC "W" number (for students) or your first and last names (for faculty/staff) and personal password. Once you've done this and successfully authenticated, you are redirected to the database.