Hamburg-Berlin-Frankfurt-Munich

Unless you are going to fly from Munich to London and have found a great price, I think I would do it as Hamburg-Berlin-Munich-Frankfurt and fly or train from there. Just for information, the Bahn has special fares starting at €49/p) from Frankfurt to London by train (uses the EuroStar through the Chunnel).

Munich is the most expensive out of all those cities, but it's still cheaper than Scandanavia, Paris, or London. You can stay at "The Tent" for 7.50 for a foam mat on a floor or 10.50 for a bunkbed. It was not the best sleep I got in my life, but it is a good deal and a fun places to stay. Be aware it's pretty far out from the city center in Munich, a 10 minute walk from the tram line that will take you there. It is near the Nymphenburg palace, though. If you eat primarily "fast food" - German style, like Bratwurst, Leberkaese, Doener Kebab, you can have very filling meals for under 6 euro anywhere in Germany including Munich. In Berlin you can eat even cheaper, Currywurst is usually like 2.50-3.50. Water and soft drinks are as much if not more as beer in all these places. A small beer at a restaurant will cost between 2.50-3, better value the bigger the beer you get, usually. German wine is also very reasonable at most establishments, usually between 3-4 euro per 0.2 liter glass. All those cities offer day tickets for public transit that are generally a good deal if you're using it a lot. A day ticket for the zones you'll be going to run between 7-12 Euro. Hostels vary, the Tent in Munich is cheaper than most. I'd budget at least 25 for hostel, you might get lucky and find something for 15-20 though. Some of the museums in these cities are cheap, others are a little bit more expensive, and there's always free things to do as well. I'd budget 15 per day for sightseeing, you might use less. Hope this helps give you an idea of the costs!

It is possible but at least for me 3 cities would be maximum to enjoy in 10 days. Otherwise it's too rushed.