" to run up an account " is another way of saying "to put it on the slate" meaning to buy things on credit at a store using their own credit card facilities or at a local shop and pay later. A lot of tradesmen run up accounts at various building stores and pay at the end of the month after their customers pay them. Examples: She was warned by her father not to run up an account at the fashion boutique because he would not pay the expensive bill any more. He had an agreement with his local corner shop to put his cigarettes and newspapers on the slate[ run up an account] and he'd pay the shopkeeper once a month.