"Character is destiny." How is this quote applicable to King Lear?
I would take the expression "character is destiny" to mean that the person who you are will in some way dictate the future events of your life. This thought does apply to some extent in King Lear. Lear himself is an emotional, eccentric yet powerful king at the beginning of the play. He desires for his daughters to flatter him with affection, even if it is false, and he is "rewarded" later on with false service from the two daughters that gave him false affection and praise. Lear's emotional side also determines future events to some extent when he is faced with the rebellion and betrayal of Regan and Goneril. Emotionally, he cannot handle what has happened and it leads to his madness.
On the other hand, Cordelia is an honorable character who speaks the truth and has a high, loving respect for her father, but her noble character only leads to her death in the end. She willingly risks her own life to attempt to save her father, but in the end both are captured. One could argue that Lear's downfall is well-deserved while Cordelia's is tragic. Their destinies are intertwined, however, and result mostly from Lear's character and actions.
Do you mean King Lear as a character? It is difficult to think that the character of anyone deserves the fate that befalls that irritable, old king that opens the play. To be sure, he misuses his power and doesn’t understand his daughters either, but his daughters subsequent treatment of him, the madness that results from that, his homelessness and suffering during the storm, and then, finally, holding his dead daughter in his arms at the end of the play—none of these seem to be his “destiny” if fortune is at all fair. Critics call King Lear Shakespeare’s cruelest play simply because his suffering exceeds his deeds, forcing us to wonder whether there is meaning in a universe that appears so unjust. And, as with Lear, Cordelia does not at all deserve her death at the end, causing many critics to wonder why in fact Shakespeare has her die for it seems so pointless in terms of the plot. As for Goneril and Regan—well, that’s another story, for there we do see them receiving their just desserts for they are greedy and cruel, and suffer accordingly