This is by no means objective research, and I haven't ever made a viral campaign. But I do follow the industry and I do participate on the Internet, so here are some elements of viral things to consider based on personal experience:
Humor
Things that are genuinely funny tend to spread. Of course humor is always debatable, but the original Old Spice viral stuff was downright hilarious if you're into smarmy and absurd humor. If it's fresh, different, and it makes people laugh, they'll tend to pass it along.
Humor's a tough gamble though. If people aren't distracted by laughter, they'll notice that you're trying to make the campaign "go viral". See the 2010 Census Payton Schlewitt campaign for the first example that comes to mind. It felt like it was trying to be funny and had a lot of components of things that go viral, so...it wasn't funny and I don't think it really achieved what it wanted to.
Linkbait
A dirty, dirty trick in my opinion, but undeniably effective. BuzzFeed built an empire on "listicles," because people seem willing to read and share things that are neatly segmented groups of ten or 20 or 40. Upworthy is building an empire on titles that make you curious enough to click (pro tip: follow @UpworthySpoilers and avoid this problem).
The broader point inside this one is that people are figuring out empirical ways to attract click and attention, and then using it over and over again. I personally believe that people will get tired of it, but in the meantime BuzzFeed and Upworthy are certainly doing well for themselves.
Nostalgia
Millenials seem to love reminiscing on the 90s, for example. I see stuff like that spread all the time. Zimbio quizzes about which "Boy Meets World" character you are or "Which Disney Princess Are You", that sort of thing.
Partisan
Plenty of stuff gets shared because it fits within a worldview of someone who feels passionately enough about it to share 20 times a day on Facebook. This is tough though, unless you touch a nerve that people really care about.
Comeuppance
Ties in with Partisan a bit. I've been seeing this one more and more. Person A thinks Group B is stupid, and when A sees a link where Group C puts Group B in its place, A loves that and wants to share it.
So, let's say you found a source where someone says that coding PHP is stupid. You make a video that is a withering retort, putting that person in his place. PHP coders see it, like it, share it, and it has your company's name attached to it. Not saying that would work, but saying that's how it could work.
A recent example that partially fits in here is Taco Bell's recent decision to use people named "Ronald McDonald" to promote their new breakfast. It got attention because it was a shot across McDonald's bow.
There's a quote going around: "While you were arguing about Rails vs Django vs Flask vs Erlang, a bunch of people shipped with PHP." What if your company had made that quote and people were discovering you as a result?
External variables
This is a different angle to consider: Virality is hard to control. Kind of like how viruses are hard to control! The Aqua Teen Hunger Force movie incident is a textbook example. Whatever the intent of the campaign was, it became something entirely different based on circumstances outside of the advertisers' control. But while people were fined and jobs were probably lost, it got national press for very little money. So, was it successful?
Placement
If you follow sites like Hacker News, Daring Fireball, or Reddit, those can be launching pads for people. Very often I'll see sites that go down when they're linked there because they weren't used to the load. Some sites like Stratechery can probably credit a few well-placed links for their blogging success (or at least the speed of their success). Not sure if this fits your definition of viral, but an organic link on Daring Fireball can end up doing a lot more good for you than an AdSense buy.
But that sort of stuff is tough to force. You need to have great content and the right people have to pick it up.
Conclusion
And perhaps that's the point of virality - it has to spread on its own, there are ultimately going to be some variables outside of your control, and for it to spread it has to be worthy for people to share without making any money by doing so. If I'm not getting paid to share a video, I'm sharing it because I find some other kind of value in it. Give a person something valuable that they would want to share.
