Finding full text of Shakespeare's works online is easy and free, since everything is in the public domain! Here's just a sample of some places to read Shakespeare's works online:
Editions and Adaptations of Shakespeare includes the complete text of eleven major editions of Shakespeare's works, from the First Folio (1623) to the Cambridge edition of 1863-6. It also includes twenty-four separate contemporary printings of individual plays, selected apocrypha and related works, and more than 100 adaptations, sequels, and burlesques from the seventeenth, eighteenth, and nineteenth centuries.