What was the name of the company of actors Burbage belonged to? |

I believe that you are probably referring to the group "The Queen's Men."  This group was formed in 1583.

As a fairly important actor of his day, James Burbage did not belong to just that one company throughout his career.  He started acting perhaps thirty years before forming the Queen's Men.  During that time, he belonged to various companies.  One of the more important of these was known as the "Earl of Leicester's Men."

However, The Queen's Men was the most famous of these.  This is largely because it was the company that performed, among other things, many of Shakespeare's plays.