If you use a recent version of Windows, the answer is simple: Just install BlueStacks, which creates an Android operating system installation within Windows, kind of like a pocket universe (Android), mostly accessible from the main universe (Windows), without having to reboot the computer. With BlueStacks, you can download and install Instagram with a few mouse clicks.
While BlueStacks lets you access Android directly from Windows, I have found some apparent limitations. To use Instagram, you need to take a picture from Instagram (I don't know if a PC webcam is compatible), or use a file on hand.
As a workaround, I would send myself an email containing images I want to use. Then I would then go to the Twitter app on BlueStacks, then I'd go to Settings, then About, then Terms of Service, which should open the default Android web browser, or give the option to do so. I could then access my email, and download the files for use in Instagram. (Update, 1/5/2013: Having Dropbox installed in both the regular operating system and within BlueStacks -- via Android Drawer .apk -- is a more efficient solution. Alternatively, if you can get it to work properly, the ES File Explorer app can provide access to your computer documents.)
Finally, I could go to the Instagram app, access any photos that have been downloaded into the Android pocket universe (operating system), crop, add a filter, then upload into my Instagram account. And yes, you can sign up for Instagram using this emulated mobile device.
Mac OS X
I recently checked, and apparently the Mac version of BlueStacks is now in beta. I figured out how to install Instagram using the alpha version of BlueStacks, which is tricky, buggy, and a bit convoluted. Anyhow, I haven't tried the beta, so here are the instructions for the alpha version of BlueStacks:
I downloaded and installed the alpha of BlueStacks onto my MacBook. Since Google Play will call you on your emulated mobile device trickery, and thus not allow you to download and install Instagram, you will have to take a chance with Android Drawer, and hopefully not download a malware-filled version of Instagram, or any other app. Here's the to-do list:
1. Attempt to install the Instagram .apk from BlueStacks.
2. Get prompted that you'll need to upgrade your Android operating system with a download from Google, which they'll allow with your fake mobile device.
3. Download and install that Google-approved upgrade, which name escapes me, but you'll get prompted by Android, so hopefully you'll be on the right track.
4. Go back to Android Drawer to re-download Instagram.
5. Attempt to install the Instagram .apk again, which should work this time.
6. Sign up or log in to Instagram from your Mac!
Edited to add, 1/4/2013: 7. Once signed in, go to the app's preferences, and turn off Instagram's Advanced Camera setting (go to Camera Settings), otherwise you might not be able to access your image files.
As mentioned many times previously, the alpha build of BlueStacks is pretty buggy, and Instagram is just marginally compatible with this Android pocket universe. Here's how to more-or-less successfully upload a photo on Instagram:
1. Instagram on BlueStacks alpha will either crash or upload your photo mirrored and upside-down.
-- and use an image program, like Preview on OS X, to flip vertically (in other words, make it an upside down mirror image), like so:
3. Now open BlueStacks.
4. Use the Twitter-browser workaround mentioned earlier (the bold paragraph) to download image files in your Android installation.
5. Open Instagram.
6. Access the Photo Gallery with the image you want to Instagram.
7. Crop, add a filter, and upload. Using the selective focus effect will probably crash the app.
8. If all goes well, you should have a not-so-instant Instagram photo, right side up and non-mirrored:
*Sorry about the hashtags in the title. It's for Twitter's benefit (or not).