The clause, "which culminates in the murder of the king", is a restrictive clause, ie it cannot be excised from the sentence without removing an essential piece of information:
In traditional grammar, restrictive clauses start with the relative pronouns that, who, whom or whose. In recent times, due to common use, some grammarians will accept restrictive clauses that begin with "which". Further, restrictive clauses are not offset by commas before the relative pronoun. So, the correct way to write this sentence would be:
Gaveston alienates the queen from the king, antagonizes the barons and embroils the country in a bloody civil war that culminates in the murder of the king.
I would personally prefer a comma between 'barons" and and", but I will leave that for now.
Just for completion, a non-restrictive clause is one which contains additional, but non-essential, information that can be excised from the sentence without affecting its intended meaning or structure. E.g If clause 4 had said:
the war started at the Northern border.
This is not essential information given the rest of the information supplied. The whole sentence would now read:
Gaveston alienates the queen from the king, antagonizes the barons and embroils the country in a bloody civil war, which started on the Northern border.