How is the play King Lear relevant even today?
Your question brought to mind the newspapers of a few weeks ago in which Kim Yong Il of North Korea announced the succession of his son, Kim Young-Nam as the new head of state to take over for him. In the same article, Kim Yong Il's brother, Kim Yong Un, was promoted by Kim Yong Il to the rank of four star general. This is a modern case of a madman who, like Lear, has an elevated opinion of himself and chooses his successor based on nepotism. Who knows, perhaps Kim Yong Il had a contest to see which of his sons loved him more -- only kidding.
To answer your question, as long as there are prideful men and women who put themselves before the good of their country and their families, the play KING LEAR will be relevant. Do you remember how they found Saddam Hussein? The former madman ruler of Iraq, who also had his family installed in all of the powerful positions in the country, was reduced to a crazed lunatic-looking man hiding in a hole in the ground. Kind of like the blind King Lear, wandering around lost.
We still have sibling rivalry today, so that issue in the play is also relevant, and we still have heirs fighting over who should get what of a parent's estate. If you read the analysis of the play here on eNotes, you will also find an excellent viewpoint that posits that this play has a very modern, nihilistic view of mankind that is relevant in today's world.