He is not in support of expanding the existing system of background checks. He is opposed to any extension of the existing background check system which, currently, only applies to FFL dealers. So yes, he is "against" background checks unless you are of the belief that the system we currently have requires them.

From his position paper

BACKGROUND CHECKS. There has been a national background check system in place since 1998. Every time a person buys a gun from a federally licensed gun dealer – which is the overwhelming majority of all gun purchases – they go through a federal background check. Study after study has shown that very few criminals are stupid enough to try and pass a background check – they get their guns from friends/family members or by stealing them. So the overwhelming majority of people who go through background checks are law-abiding gun owners. When the system was created, gun owners were promised that it would be instant, accurate and fair. Unfortunately, that isn’t the case today. Too many states are failing to put criminal and mental health records into the system – and it should go without saying that a system’s only going to be as effective as the records that are put into it. What we need to do is fix the system we have and make it work as intended. What we don’t need to do is expand a broken system.