How could the American Revolution be fought in the name of independence while also existing as a collection of slave owning colonies? How did early colonial leaders justify slavery in the midst of fighting a war to support personal liberties?

Freedom can be defined in many ways. The political sovereignty or freedom of the colonies from British control did not, as the question implies, amount to freedom from slavery for African-Americans in the colonies. Many U.S. history texts do a good job of confronting this issue. Eric Foner's Give Me Liberty: An American History deals directly and explicitly with the captivity of slaves in the context of the American Revolution.
The historical evidence suggests that the paradox was not lost on the British or on the American revolutionaries. The British brought the contradiction to light for propaganda purposes and, with the promise of freedom, many slaves chose to fight on the side of the British during the Revolutionary War. Many others fled behind British lines or emigrated to Canada. The colonial governments sought to combat this trend by passing manumission laws, which allowed slaveowners to free their slaves. As a result, some slaves fought on the side of the colonists with the expectation of improved social status or freedom.
After the colonial victory, a combination of political compromise, Southern interests, economic motive, and racism kept the institution of slavery intact.