Do you want to get the cheapest business class airfare possible?
You would simply need to sign up for the e-newsletters and fare alerts provided by the airlines to watch for us to take advantage of the flash sales or online fare / price web promotions offered only by the airline.
Depending on the airline you travel on, you could use your credit card points or frequent flyer (mileage) to change to the premium cabin, especially if you noted that the airline had discounted seats open.
You could find a travel agency that can act on your behalf to book your business class airfare within the discounted business class airfare category (it is a hack to use a travel agency in this sense because they won't earn recovery like other travel management companies).
You can keep an eye out for mistake fares (which is when an airline or a travel agency misprints a pricing mistake) posted on deal sites like Scott's Cheap Flights or Secret Flying as they also will often have cheaper fare options in terms of business class flight paths that relate to their example but are confusing even.