Everyone's different, but that's a lot of cities in ~30 days. Keep in mind that air travel in China is not terribly efficient, in the sense that you need to arrive early, and expect to arrive late. You could wipe out most of a day each time you go to the airport, and with 10 cities that means 1/3 of your time wiped for air travel. True, you probably can arrange to use part of that day upon arrival (or departure), but you can't count on it. Also, while you seem to have the cities lined up in a least-distance direction, you may find it is more difficult to find non-stop flights on those city pairs than if you you took a different route. All you can do is check flight schedules to try to minimize connecting flights (e.g., flights to/from Jiuzhaigou are Chengdu, Chongqing, Xi'an and Beijing, I believe, so ideally you'd go to/from one of those cities). That leaves you with 2 days/city, which is awfully tight when some of those places may require additional time to get to from the closest airport.

As for cost, it all depends on how you want to travel. Depending on airfares, you could be looking at £150-250/flight, although you can do better. That's 1/3 to 1/2 your budget. Do you want deluxe accommodations? Food should be less expensive, but don't forget admissions fees and such, which can add up. But certainly doable for that budget.

Personally, if it were me I would probably visit somewhat fewer cities/places and spend more time there. 30 days is a long time to go-go-go. Depending on your interests, I would tend to drop some of the southern cities, and save those for a separate trip (i.e., Shenzhen, Sanya, Guangzhou). If you stick with your order you could go from Zhangjiajie to Lijiang and spend a number of days in Yunnan, then Chengdu and Jiuzhaigou. That would give you at least 3 days per "stop" (although you might adjust--e.g., you might stay only two nights and one day in Jiuzhaigou, but more in Yunnan).