For small loads, a milk crate bungie-corded to your rear cargo rack works really well. My crate never leaves my bike. Perfect for a full bag of groceries, a load of library books, takeout food, etc.
For larger loads, I also recommend a trailer. I often use my child's trailer for cargo, but in your case, you could grab a proper cargo trailer and attach the rubbermaid bin. I've seen these for sale, and some people also seem to Frankenstein them together.