North Cascades National Park is located about 3 hours Northeast of Seattle and 3 hours Southeast of Vancouver, BC. You can easily visit it when visiting either city. The nearest airports are going to be in Seattle or Vancouver.
My recommendation would be to fly into Vancouver and explore Vancouver for a few days. Make the drive down and spend a day in North Cascades National Park. After you have got your fill of the beautiful mountains and emerald green lakes, drive down to Seattle and explore this city. We did a similar trip and had a lot of fun.