If you want to record video (for this, or whatever else) I recommend using OBS, or Open Broadcaster Software. It's a little difficult to set up properly, but it's completely free, and you can easily find tutorials on it.
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Just upload the .mp4 file to any video uploading website. Unfortunately, I don't have much experience with uploading content, but I'd assume YouTube would be pretty simple to get started on (well, assuming you already have a YouTube account, which, if you're already associated with Google, isn't an issue).
What you're looking for shouldn't be hard to find. I just Googled "mp4 to gif" and came up with a ton of results, like http://ezgif.com/video-to-gif, http://www.zamzar.com/convert/mp4-to-gif/, https://convertio.co/mp4-gif/, and more.