Where you put your test code depends on what you intend to do with the code. If it's a stand-alone class that, for example, you intend to make available to others for download and use, then the test code should be a project within the solution. The test code would, in addition to providing verification that the class was doing what you wanted it to do, provide an example for users of your class, so it should be well-documented and extremely clear.
If, on the other hand, your class is part of a library or DLL, and is intended to work only within the ecosystem of that library or DLL, then there should be a test program or framework that exercises the DLL as an entity. Code coverage tools will demonstrate that the test code is actually exercising the code. In my experience, these test programs are, like the single class program, built as a project within the solution that builds the DLL or library.
Note that in both of the above cases, the test project is not built as part of the standard build process. You have to build it specifically.
Finally, if your class is to be part of a larger project, your test code should become a part of whatever framework or process flow has been defined for your greater team. On my current project, for example, developer unit tests are maintained in a separate source control tree that has a structure parallel to that of the shipping code. Unit tests are required to pass code review by both the development and test team. During the build process (every other day right now), we build the shipping code, then the unit tests, then the QA test code set. Unit tests are run before the QA code and all must pass. This is pretty much a smoke test to make sure that we haven't broken the lowest level of functionality. Unit tests are required to generate a failure report and exit with a negative status code. Our processes are probably more formal than many, though.