If you're planning on using the animation solely on the web, it might be better to go with a format that's more web-friendly than video, such as Javascript or CSS animations, or animated svg.
Upside is that these are usually lighter in weight, and thus faster to load, downside is that these can't always achieve the 'coolness' of video effects.
1. javascript
A first option for Javascript animations is coding it yourself using pure javascript, jQuery or an animation framework like GSAP. This gives you most control and flexibility, but you'll need to get your hands dirty.
A second option for Javascript animations is Edge Animate, an Adobe tool specifically made for animations on the web. Quite easy to use, no need to know a lot of code, and a lot of great tutorials available on the web. Downside is that EA produces a lot of files, which can make it hard to integrate into a website if you don't know any code.
2. css
CSS animations require you to know a bit of code, but are powerful, and easy once you get the hang of it. Upside: no need to use excessive javascript, which will keep your website payload down. Downside: not as powerful and flexible as javascript animations.
3. animated svg
SVG is a vector graphics format, so best suited for vectorial images (since you're using PS, I'm assuming that might be a problem). The upside of using an animated svg is that you can save it all to one file and just put it in your HTML like any other image. Downside: IE isn't playing along nicely.