First thing, a credit application will be taken for a small fee of $25-40 per person. It will ask for your work history, rent history, social security number if you have one and annual income. It will also check your background for any felony record.
If you just moved here, you will need to provide a letter proving you have been hired by a company and your start date.
If all that passes, most landlords or leasing companies for apartments ask for 1st, last month's rent and a deposit with some of that non-refundable.
What month do you plan on moving out to Seattle?