How do I run GUI version of NetworkManager on DWM?

Provided that your system includes the network-manager-gnome package (and its dependencies), you should be able to run NetworkManager's GUI connection editor from the command line using

nm-connection-editor

You can attempt to start the GUI desktop applet as nm-applet, however it may not be able to interact with / dock within a simple WM.

You need a system tray for the tray icon you want (nm-applet), which DWM doesn't provide by default. Their are a number of stand-alone tray programs for light weight window managers like DWM. You could try the venerable stalonetray or systray, the latter of which is by the same author as DWM.