The Hero's Journey is a monomyth applied to many cultures and types of literature. Author and mythology expert Joseph Campbell offers a detailed explanation of the steps of the journey and many examples that exist in different cultures. The Hero's Journey has similar features across cultures. The steps include the call to adventure, refusal of the call, belly of the whale, and others (see link). The journey is one towards self-actualization and fulfillment of divine or human purpose, and is usually fraught with danger and many challenges.
In King Lear, the main characters that experience the journey are Lear himself, and his daughter Cordelia. Both characters experience an opportunity to embrace a challenge or journey. Cordelia is asked to speak for herself and demonstrate her love for her father, but, knowing such a statement might indicate her desire for her father's wealth, she chooses to say nothing. This causes her sisters to work in league against her to curry their father's favor. Lear's challenge is to reveal the loyalty and integrity of his daughters. In the end, Lear finally realizes Cordelia is the most loyal, and in death her purity and love are immortalized.