Slavery did not violate the actual letter of the Constitution. This is particularly true if we look only at the original Constitution (and not the Bill of Rights). However, it clearly does violate the spirit of that document.
The Constitution does not condemn slavery. It does not explicitly mention slavery. However, the people who wrote it clearly accepted slavery and believed that slavery was compatible with the Constitution. Today, however, with our modern ideas about race, we no longer think that slavery was compatible with the Constitution.
In order to say this, we have to say that the Constitution was meant to provide liberty and freedom to all Americans. We can clearly see in the Bill of Rights that Americans were meant to have certain freedoms. We can see from the original Constitution that the Framers were very interested in ensuring that the government could not abuse the rights of the people. For example, they set up a system of checks and balances to prevent the government from becoming abusive. If the Constitution is set up to ensure the rights of the people, then slavery is (at least in our modern eyes) contrary to the principles of the document.
The Constitution established principles of individual freedom, " Life, Liberty, and the Pursuit of Happiness." Of course if you're a slave, you have no freedom and you're not treated like an individual.