Hey,
your plans sound reasonable and can definitely be realised. I think 50 pounds (converts to around 58 Euro) a day should be enough per day considering that most hostels cost around 20 Euro per night including breakfast.
It might be worth getting a Youth Hostel Pass (http://www.jugendherberge.de/en/ and http://www.jugendherberge.de/en/basics/membership/index.jsp), I paid 12 Euro last time (I'm under 26). Once you've got such a card, you can stay at the hostels of the German Youth Hostel organisation, they are usually in a really good shape, centrally located and not overly expensive (from around 17 Euro per night incl. breakfast, e.g. this one in Hamburg (http://www.jugendherberge.de/en/hostels/search/portrait/jh.jsp?IDJH=522) charges 19.60 per night incl. linen and breakfast). The membership cards can be bought at the hostel and are valid for a calendar year.
Apart from that, the Wombat chain is supposed to be really good. I just saw that they're only in Berlin and Munich though.
Frankfurt isn't really worth spending three days there as there is not that much to see, I recommend visiting Marburg (http://en.wikipedia.org/wiki/Marburg) for a day while you're in Frankfurt, it is a beautiful medieval town with a castle, an old university (founded in 1527, the Brothers Grimm were there for a while), the 800-year-old Elisabeth church (http://en.wikipedia.org/wiki/Elisabeth_Church_(Marburg) which still attracts pilgrims and lots of half-timbered houses. It is a university town and has the highest pub density in Germany - lots of student parties and pubs are packed every night of the week. It's an hour from Frankfurt by train.
Weimar (http://en.wikipedia.org/wiki/Weimar), the home town of Goethe and Schiller and founding place of the Weimar Republic, is not too far away from Leipzig and has a lot of history, especially concerning Germany's "poets and thinkers" and is a beautiful town as well. The former concentration camp Buchenwald (http://en.wikipedia.org/wiki/Buchenwald) is nearby, in case you are interested in the darker episodes of German history. It has a museum attached and gives you a really humbling idea of the living conditions for the prisoners and of the horrible things that happened there and all over Germany in the 40s.
Also, Eisenach with the Wartburg (http://en.wikipedia.org/wiki/Wartburg), the castle where Martin Luther found refuge, is worth a visit.
Münster is also really nice and might be worth checking out on your last leg - Cologne to Frankfurt.
When you're in Hamburg, you could do a day trip to the North or Baltic Sea. Then again, Hamburg probably has a lot more to offer than you can see in three days so maybe you should do that next time you come.
Hope this helps, if you have more questions, feel free to contact me.
butafly
[ Edit: Edited on Nov 14, 2008, at 5:09 AM by butafly ]