Shakespeare's style of writing was principally "blank verse," a form of unrhymed iambic pentameter. Like much Elizabethan literature, his plays were typically replete with figurative language and rhetoric.
Rhyming couplets appeared occasionally in Shakespeare's work, but they were not characteristic of it.
Far more important to Shakespeare's writing style was the steady rhythm. "Iambic pentameter" refers to the technique of following an unstressed syllable with a stressed one (this pairing being called an "iamb" or "iambic foot") five times on each line ("pentameter").
It has been suggested that Shakespeare's choice of words sometimes seem unusual because they were carefully chosen to fit this metrical pattern.