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.