Do you mean the name of the King in his plays, or the name of the ruling monarch during his lifetime?
If the former, he wrote several plays about Kings- you need to specify which play you are talking about. If the latter, for most of his life it was a QUEEN that was on the throne at the time, Elizabeth 1st. He lived into the reign of her successor James 1st for 13 years, passing away on 23rd April 1616, his birthday and also St. George's Day.
The Lord Chamberlain's Men became The King's Men.
In 1603, The Lord Chamberlain's Men became The King's Men.
It indicates (as the names of all Elizabethan companies did) the name of the patron of the company.
King Lear does, along with his fool.
Shakespeare was a shareholder in Lord Hunsdon's Men who changed its name to the Lord Chamberlain's Men, who later became the King's Men.
Hamnet
the Globe
A Midsummernights Dream
Queen Elizabeth l
Ophelia
Pusssy
Kurosawa's Ran.