Tokyo Metropolitan Government Building Observatory Deck is free and gives a good 360 view from the west of the city including great views on Mount Fuji on a clear day. Tokyo Tower gives an alternative perspective from the center of the city but it's a little pricier that it could be to get to the top. Tokyo Sky Tree is one of the world's tallest buildings offering phenomenal views and has a interesting shopping center and restaurants at the base. It's at the East of the city and good to combine with a trip to Akasaka.
If you are interesting in a bar with a view the rooftop bar on 52F of the new Andaz Hotel is very impressive.
(Note the Roppongi Hills City View and Skydeck - impressive outdoor view from 250m - mentioned by others are closed until April 29 2015, and they Sky Deck is always closed during bad weather).
The Tokyo Metropolitan Government Office in Shinjuku. It's for free, unless you'd want to buy some souvenirs. You will take the elevator up to the 45th floor where you will get a good 360-degree view of the whole Tokyo area and even Yokohama. You will also see Mt. Fuji.
I agree with Andrew's recommendation of the Tokyo Metropolitan Building. The only downside is that there can be long queues for the lifts, but I think the fact that it's free makes it worth the wait. On a clear day the views are amazing - I had a wonderful view of Fuji when I went up there. There are also souvenir shops and a small cafe at the top, so you can tick off several things at once!