Update VS2015. https://msdn.microsoft.com/en-us/library/mt752379.aspx is Update 3. You can use VS2017 if you have Intel Parallel Studio XE 2017 Update 4.
The first error "error opening the compiled module file" is critical - you left off the rest of the message so I don't know which module it wanted. Adding all the .f90 files to the project should be sufficient.
gib>>You created your project as a Console Application. You need to create a new project as a Library, and select DLL.
JohnC>>I do that.project built successfully but I got this error...Visual studio cannot debug because a debug target has not been specified.
You cannot run a DLL directly, thus no startup project.
Your Visual Studio Solution can have multiple Projects. In your case, for starters, you need a small test console application Project, plus your new DLL Project. The small test Project is to be set dependent upon the DLL project. Your test console project makes test calls to your DLL.
In MS VS you will need to specify the Startup Project as the console application (Right-Click on the Test Project in the Solution Explorer pane and click on Select Startup Project.
As far as I am aware the issues around accessing a DLL from a C# program are not Fortran issues. It is possible that you need to tell the C# linker about the .lib file that was created together with the DLL.