I don't think there's a secure way of doing this. No matter what kind of system-unique, or user-unique named object you use - malicious 3rd party software can still use the exact same name and that would prevent your application from starting at all.
If you use the method of checking the currently executing processes, and checking if no executable with the same name is running - you'd run into problems, if the malicious software has the same executable name. If you also check the path, of that executable - then it would be possible to run two copies of your app from different locations.
If you create/delete a file when starting/finishing - that might be tricked as well.
The only thing that comes to my mind is you may be able to achieve the desired effect by putting all the logic of your app into a COM object, and then have a GUI application interact with it through COM interfaces. This would, only ensure, that there is only one COM object - you would be able to run as many GUI clients as you want. Note, that I'm not suggesting this as a bulletproof method - it may have it's own holes (for example - someone could make your GUI client to connect to a 3rd party COM object, by simply editing the registry).
So, the short answer - there is no truly secure way of doing this.