So, a hate campaign is a campaign of hate. It can be qualified: a political campaign of hate, AKA, a political hate campaign. That would mean an individual or organization or politician involved in spreading "political hate" towards some group. A nasty business, to be sure. The political refers to the field of endeavor (the political arena) and not the person necessarily, i.e., the politician.
Please note: A campaign can be carried out by a politician running for office (BrE: standing for office) or any other individual or group. There are a lot of social media campaigns in favor of this or that or against this or that.