To Obtain a Monster Contract you can talk to any Monster Contract Dealers in the world. As of this entry there is only one dealer available.
Rio The Contract Dealer
Outside New Player Building
When you talk to them you will receive a Gump offering you a contract. Click Okay to be given a contract, or cancel to refuse. You will have a wait timer on every character of 1 hour after obtaining a contract. This timer will wear off as well by turning in a completed contract.
To complete the contract you must kill the required amount of the required monster on the contract. It will auto-add the kill to your contract as long as it is in your main backpack on the kill. This also will only add to one contract at a time.
Return to Rio after completion and drop the contract onto him to receive your bank check reward.
You can also obtain a Monster Contract Book for the Contracts by crafting in the Inscription menu. You can add up to 500 monster contracts into a book. They act the same as a bulk order book or a power scroll book. You can add prices and sell contracts on your vendor as well through the book. You will need a Monster Contract Recipe to make these books which you can obtain in a Metal Detecting Chest.