This video will have taken a very long time to make. Autotuning video takes such a long time because, as far as I'm aware, there isn't any good video-editing software out there which allows you to autotune audio in-app. Your best bet would be to mix the audio by itself and then cut up the video to match it. It's annoying and painful but it'll produce a great-sounding track if done meticulously.
There is plenty of good software out there if you want to start on autotune - from paid-for examples like Melodyne, Ableton, Logic Pro, or Pro Tools to free examples like Audacity or GarageBand. Note that you're unlikely to get a very good autotuner without paying for the software; as far as I know, you can't selectively change the pitch of individual notes on things like Audacity or GarageBand... however, you might be able to find free trials of the paid-for software (for example, Ableton has a 30-day Free Trial here).
Here are the steps I would take to make one of these videos:
- Import both your intended track (Havana in this case) and your video into your video editor
- Cut your video to match the rhythm of your track. With your example video, you would cut the track to match the rhythm of Havana (ie without auto-tuning it)
- Find a way to extract the sound from each mini clip and extract all of your samples into separate files in a folder to keep them grouped up
- Import that folder into your music editor of choice (or mass-import the individual files)
- Match the files up correctly like they are in your now-chopped-up-video
- Now that the samples are separated, you can easily select each pitch and change it to the desired note
- Once all the audio is edited (tuned, mixed etc), export the track as an audio file
- The audio track should match the video now because you cut them up together initially. If they don't, you can tweak it just to make it perfect. Just in case the tracks don't match up, I would avoid saving your chopped-up audio with your intended track - this will mean that your video will have two audio tracks (the cut-up one and the actual song), and you can splice the audio of your cut-up track to warp the timing a bit without affecting the original and putting the song out of time.
It's a long process but being methodical about it will save you a lot of time.