What would be the best way to write Objective-C on the Windows platform?
If you just want Objective-C but not any of the Cocoa frameworks, than gcc will work on any platform. You can use it through Cygwin or get MinGW. However, if you want the Cocoa frameworks, or at least a reasonable subset of them, than GNUStep and Cocotron are your best bets.
Cocotron implements a lot of stuff that GNUStep does not, such as CoreGraphics and CoreData, though I can't vouch for how complete their implementation is on a specific framework. Their aim is to keep Cocotron up to date with the latest version of OS X so that any viable OS X program can run on Windows. Because GNUStep typically uses the latest version of gcc, they also add in support for Objective-C++ and a lot of the Objective-C 2.0 features.
I haven’t tested those features with GNUStep, but if you use a sufficiently new version of gcc, you might be able to use them. I was not able to use Objective-C++ with GNUStep a few years ago. However, GNUStep does compile from just about any platform. Cocotron is a very mac-centric project. Although it is probably possible to compile it on other platforms, it comes XCode project files, not makefiles, so you can only compile its frameworks out of the box on OS X. It also comes with instructions on compiling Windows apps on XCode, but not any other platform. Basically, it’s probably possible to set up a Windows development environment for Cocotron, but it’s not as easy as setting one up for GNUStep, and you’ll be on your own, so GNUStep is definitely the way to go if you’re developing on Windows as opposed to just for Windows.
For what it’s worth, Cocotron is licensed under the MIT license, and GNUStep is licensed under the LGPL.