In this context, "ticking against" and "tick from" are referring to marking something as complete using a checkmark. When you tick against something, you are essentially putting a "tick" next to it to make sure that it is done. The person is marking things down on a ledger (a book that contains information about financial accounts). The word "tick" is not used a lot in American English, but it is commonly used in British English. I hope this helps. :)