This is one easy method without having to rely on third party applications.
This method currently works on any Windows 10 machine. It also works on Mac, by substituting "Command" for "Control".
There are limitations to this method. This will only allow for you to upload a single image into one post. Plus, a few of the Instagram features are unavailable. If I find a work-around, I'll update this here in the future.
Steps to upload from computer to Instagram
1. On your computer, go to your Instagram profile. Click Shift + Ctrl + i
Your screen will then look like this.
2. Click Reload Page (refresh screen) from the browser. The change you'll notice is the + sign at the bottom-center now. This is what you'll click to upload your image.
3. Click on Close in the Developer module (the X on the developer module to close out that feature and to expand your profile). Now, there's only Instagram on the screen.
4. Click on the + at the bottom. This will open your folder options to select which image you wish to upload. Again, you won't be able to select multiple images.
5. Scroll down and Click on the Double Arrows at the lower left corner of the Instagram control panel to fit your original image to the screen.
6. Click Next in the upper right of the screen in the Instagram panel.
7. Complete filling in the usual details for your Instagram post then click Share.
After posting, close out your Instagram tab to return Instagram to normal for future use.
Also, you may want to check your post on your phone to see if you need any further editing to be done. Sometimes text formatting in the caption will appear different once you post.