Seattle-Tacoma International Airport offers a range of connections to the University of Washington Campus.
The Link light rail is an easy, great way to get to the UW from the airport. It picks you up at the airport, comes about every 5-15 minutes, and drops you off right on the UW campus; just a 15 minute walk from the dorms you will be staying in.
From the arrivals hall at Sea-Tac Airport, follow the signs for Link Light Rail; the station is a short walk from the terminal on the far side of the parking garage. Tickets must be purchased at the station before boarding. The cost is $2.50 and can be paid in cash or with a credit/debit card. Take the train to the last stop (UW). Here are directions walking from the UW link stop to the Lander Hall where you will check in.
Taxis and uber/lyft are also available from the airport. Note that for uber/lyft you will meet the car in the 3rd floor parking lot for pick up.