"We now have a need to go with 10G connections from the backup servers to the core and we are wondering if we should go with the 10G connections to Core2 or Core1?"

Answer depends much on where the server ports are located that will be the other side of the backup traffic. If both are on the same chassis, then you have 6500 chassis bandwidth to handle the load, otherwise you also need to concern yourself with bandwidth between the two 6500s.

"We are concerned because the load on Core1 gets heavy at times and was wondering if putting the 10G on Core2 will help any."

Don't see how it will help if you have to move backup traffic between the two 6500s in addition to ports on each 6500. If backups ports can be grouped on one chassis and routine ports on the other, that might distribute the load better across both.

Assuming you're already using sup720s, you might want to insure you're using appropriate line card(s) for the bandwidth loads and perhaps use DFCs (if not already).