Was William Henry Harrison a Democrat?

No, William Henry Harrison was a Whig, who defeated his Democratic opponent, the incumbent Martin Van Buren in the 1840 presidential election. The Whig Party were the political descendants of the Federalists, the party of President John Adams and Treasury Secretary Alexander Hamilton. They advocated a strong central government, the development of a modern industrialized economy supported by a federal banking system, and a strictly constructivist interpretation of the Constitution to reign in the excesses of government.
Whatever platform Harrison may have been elected on, it didn't much matter, as he ended up serving only thirty-one days in office, the shortest presidential term on record. It's believed that Harrison caught typhoid fever on the day of his inauguration. Apparently, this was due to his not wearing a hat or overcoat on that cold, wet day.