And another echo....from the Albert Dock and out-and-back to Otterspool is your best bet. I'd do it early rather than after dark though as there are some sections where you'd feel unsafe in the dark.
South Liverpool is very green and leafy and there are quite a few parks you can take in, if you've ventured out of the city centre: Sefton Park, Calderstones Park, Princes Park, the golf club at Woolton....etc. etc.
If you're running in the early evening, you'll usually find other runners doing circuits of the road that loops around the outside of Sefton Park (but watch out for tree stumps on some sections).
This walkjogrun route shows you how you'd get there running from the Albert Dock. It's about 2 miles or so around the outside of the park. Not my route btw but if you plug "L1" into walkjogrun.net, you'll get a few coming up that start from the Albert Dock.
http://www.walkjogrun.net/routes/current_route.cfm?rid=0AB1F986-AE5B-5703-BD22779BAA6C0FC2
ps. There is (or used to be) a private gym on the Albert Dock. Your hotel should be able to tell you the name of it. There are also some hotels there (like the Holiday Inn and the Hilton) that have gyms you might be able to use. If you happen to be a Greens member, Greens Liverpool is about 10 minutes jog from the dock and easily accessible from Brunswick station on the Merseyrail network. http://www.greensonline.co.uk/club_liverpool/index.php#0
Edited: 15/09/2010 at 17:01