DaveParillo's answer actually has nothing to do with the question. The IBM Lotus Notes/Domino 8.5 Forum thread linked to deals with the application not launching at all.
What Toby is saying is that Notes launches just fine, but after it has been hidden or pushed to the background by another application, clicking the dock icon doesn't bring Notes to the foreground.
That said, I had the exact same problem, and finally solved it. Previously, I had Notes installed in my personal Applications folder (i.e., /Users/blair/Applications). When I upgraded to 8.5.1, Notes installed itself to the default location in the system Applications folder (i.e., /Applications). Instead of putting the new version in my personal Applications folder and deleting the old version, I simply deleted the old version and left the new one right where it was. This, apparently, caused the problem. Today, I moved Notes into my personal Applications folder, and it works just fine.
In short, always keep Notes in the exact same location as it was when you launched it for the very first time, after the very first install.