Start Windows service automatically after installing product

In your installer class of the service itself, use the event AfterInstall to start the service. After setup has finished, the service will start.

public ProjectInstaller()
        {
            InitializeComponent();

            AfterInstall += new InstallEventHandler(ProjectInstaller_AfterInstall);
        }

        void ProjectInstaller_AfterInstall(object sender, InstallEventArgs e)
        {
            System.ServiceProcess.ServiceController sc = new System.ServiceProcess.ServiceController(serviceInstaller.ServiceName);
            sc.Start();
        }

Also have a look at this Source.