This tutorial explain how you can upload an Insta Story through you desktop/laptop computer (Win/Mac/Linux) without the need of your smartphone.
First you have to Install: Websta for Instagram
To be able to upload an photo to an Insta Story your screen must be in Portrait mode in the moment of the upload. Before and after the upload the display can be in regular (Landscape) orientation
If you have 2 monitors is more easier because on of the monitors can be in Portrait mode and the oder in Landscape mode. This also works with only 1 monitor.
See this youtube Tutorial that explains all the process:
This image gallery show the Portrait/Lanscape mode configuration on Windows/Intel/AMD/Nvidia. This works on Win/Mac/Linux, just adjust the orientation settings.
UPDATE: It is also possible to create Keyboard shortcut to in a easy and fast way upload the Insta Story as Explained here: http://www.intowindows.com/how-to-create-keyboard-shortcut-to-rotate-screen-in-windows-10/
After and before upload you don’t need to have the screen in Portrait mode. We only need it in the moment of uploading the picture. That takes 15 seconds, so this keyboard shortcuts helps doing that step
Example for Intel Hotkeys