WASHINGTON—Democrat Hillary Clinton is launching her general election campaign in earnest this week with a speech Monday in the battleground state of Ohio and a new television ad that casts her as a leader who can bring the nation together, while Republican Donald Trump pits people against one another.

The TV spot, which will begin airing on Thursday, opens with an image of the Statue of Liberty, as Mrs. Clinton says “we face a choice about who we are as a nation.” It then cuts to footage of Mr. Trump from a February rally...