The best view of Seattle undoubtedly incorporates the skyline of the downtown area. It's often nice to have the skyline rising above a body of water, which is typically the Puget Sound, but can be Lake Union. There are a number of different locations throughout the city that provide the vantage point to get the best view of Seattle, whether that be on a ferry ride out to one of the islands in the Puget Sound, or one of the various Seattle Parks on the waterfront.
Kerry Park is by far the most popular spot to capture a view of Seattle. It's conveniently located in the Queen Anne neighborhood of Seattle, a short distance from downtown, and in addition to the skyline, provides front and center views of the Space Needle. It's the perfect place to snap a selfie, and expect to find a large number of tourists doing the same, as this is a popular stop in Seattle. While I would argue it's not the best view of Seattle, as it's difficult to capture the Puget Sound and the skyline together in a photo, the convenience more than makes up for it, and it's a very scenic view.