Having written a timeline on this very premise, IMO there are two things you need:
1. Weaken the unity of the Democratic Party
2. Improve the unity of the Whig Party
In my timeline I had the 1835 assassination attempt on Andrew Jackson succeed. This resulted in Martin Van Buren becoming President 2 years earlier than IOTL, but because this was such a new phenomena, he wasn't nearly as strong in his position as others in the same position IOTL. This lead to a stronger North-South fracture within the Democratic Party. On the other hand, you need the Whig to unify on the issue of slavery. If you have unify against slavery you can have many of the Northern Abolitionist Democrats jump ship to the Whigs, while the Democratic Party becomes forever associated with Slavery and Treason (assuming a civil war still occurs).