The best way to revise this sentence would be by placing the line "who ran in circles around the circumference of the tree" between commas.
This line is acting as an adjectival subordinate clause. It is modifying the sentence's subject -The squirrel- and is introduced by a relative pronoun that is working as a subject.
Therefore the correct way to write this sentence would be:
The squirrel, who ran in circles around the circumference of the tree, chattered angrily at the curious cat.