Poet Laureate Carol Ann Duffy composed the work for Thursday's ceremony at Leicester Cathedral.
Cumberbatch, who will play the king in a BBC Two drama, has been identified as a distant cousin, Leicester University said.
Richard's remains will be laid to rest on Thursday and his tomb inside the cathedral unveiled on Friday.
The poem, simply called Richard, is described as a "meditation on the impact of his finding and the legacy of his story".
The cathedral said Cumberbatch is "looking forward" to the service and has a "passion to know more" about the king because of his role in the drama.
Ms Duffy said: "It is a privilege to be involved, in a small way, in this unique event and to have seen the style and grace with which the city of Leicester has made history alive."
The 14-line poem includes the phrase "grant me the carving of my name" which will finally be achieved with the unveiling of his tomb on Friday.
The University of Leicester revealed Cumberbatch was a third cousin, 16 times removed of the Plantagenet king.
Since Sunday, the city has been marking his reburial and thousands of people have turned out to view his coffin.
Richard, the last English king to die in battle, was killed at Bosworth Field in 1485, at the end of the Wars of the Roses.
His skeleton was found under a car park in Leicester in 2012.
Actor Benedict Cumberbatch is to read a specially written poem at the reburial service for King Richard III.