You're going to have problems.
The cheapest Android phone I can find in my brief search, new, is the HTC Dream (T-Mobile G1) for ?145 from Amazon. The Dream was the first (and is the hence the oldest) Android phone, so likely to be the cheapest. It's stuck on Android 1.6, but is still a capable phone, and it has a reasonable keyboard. It doesn't have a lot of internal storage though, but if you root it this won't be a problem.
The only alternative with a keyboard would be the Motorola Milestone (?350 B-grade, which means it's been returned by someone and checked over, from expansys.com). But as you can see, this is getting way above ?100 already. The Milestone also comes with Android 2.1, which is much better than 1.6. Android 2.1 is the latest official release.
There's always eBay of course, if you want cheaper.
Both the HTC Dream and Motorola Milestone are rootable. The Dream has a load of ROMs for it, but most notable it can run Cyanogen MOD, which is by far the most popular custom ROM.
Powerful? Well - you pay for what you get. The Dream is not the slowest Android phone, but it is the oldest. It has a circa 528MHz processor, the Milestone has a 550MHz processor of a more powerful type. Current Android phones nearly all have 1GHz processors.
Easy to get? Both are. Amazon has the HTC Dream, expansys.com has the Motorola Milestone.