"HTML5 compatibility" is a very vague thing.
When people ask about HTML5 compatibility, they generally mean "what browsers support these new-ish browser features X, Y and Z which I want to use?"
There are a whole raft of features which have been added to browsers in the last couple of years, and which are now commonly referred to as "HTML5".
In fact, there aren't any browsers which support every new feature out there.
What you need to do is work out which features have wide enough support to make them worth using, which features you'd like to use but are happy to work around if you encounter a browser that doesn't support them, and which features you absolutely have to use to achieve what you want to do.
A fairly comprehensive list of new browser features, along with browser support charts for them all is available at http://caniuse.com/ (if you scroll to the bottom, you'll see in the overall compatibility table that the very best current browsers only support 89% of features they've tested. This will improve over time as new versions are released... but of course, also new features will be introduced too)
For determining at run-time whether the user's browser supports a given feature, you can use Modernizr. This is a Javascript-based tool which will give you a set of CSS classes and Javascript flags which tell you what features are supported. You can use this to trigger alternate behaviour in your site if the browser doesn't support a feature you want. (Modernizr also includes the HTML5Shim functionality, which allows IE to at least cope with HTML pages containing new HTML5 elements).
For more cross-browser compatibility, there are a whole range of hacks which have been written to allow older browsers (mainly IE to be fair) to support a range of newer features. You can see a fairly comprehensive list of them here: https://github.com/Modernizr/Modernizr/wiki/HTML5-Cross-browser-Polyfills
Obviously, trying to run more than a few of these at once in IE will severely impare your site's performance, but it can be handy if you need to support one or two features. My favourite at the moment is CSS3Pie, which gives IE6/7/8 support for CSS rounded corners, shadows and gradients.
Hope that helps.