Currently I have 4 solutions that are independent projects, however there is quite a bit of duplicated code when it comes to a few areas of the applications.
At the moment it is simply a few forms and their associated code.
When I want to change or improve something, I have to copy and paste to all relevant projects.
I looked at creating a new project within one of the solutions for the .dll/class library, but I felt that this was incorrect. (Please say if I am wrong).
As it is a component for all the applications, I decided to create a new solution for the .dll/class library and am looking at moving the shared code over to that - but, having never gone down this route before, what are my options from here?
Am I able to then include this solution within the others if I need to make a simple change and have it updated in all the projects or instead, should I always be working on the shared component in a separate instance of Visual Studio, outside of the applications using it?
That's exactly the right way to handle this situation.
You can include projects in multiple solutions by right-clicking the solution and selecting Add Existing Project...
Any changes you then make will appear in all solutions. The only problem this leads to is that it's possible to break one solution from another. This is where automated builds on commit to source control come into their own.