Tuition is covered by the department or individual labs, except for approximately $253 in student fees each quarter, which are your responsibility. Your initial deposit will be credited against the fees you pay for Autumn Quarter. Summer Quarter student fees are lower (approximately $130), since you register for fewer credits.
Your main tuition balance should be resolved by the second or third week of the quarter. It is (way) less automatic than one might hope - please don't be alarmed when you first see that huge tuition balance.
Please be sure to pay your student fees by the date shown on the UW academic calendar, typically by the end of the third week of the quarter.