Hard to choose the best viewpoint in Lisbon, but São Pedro de Alcantara is definitely one of the top ones. It's like sitting on a terrace with a view of the eastern side of the city. There is a kiosk where you can get some refreshments.
You can see it from many points in town - it's magnificently raised on the hill next to the fortress. When you get to the top where it stands you can see whole city.
We went there by car in some kind of a tour with other couple and a driver. It took us less than 3 hours from the parking lot, the weather was warm and sunny but it was necessary to cover the body as the sun was strong nonetheless. Pleasant day, not many people hiking at that time (we started before 7am).