I would always start with asking for work. Whether or not it should be this way, my experience has shown that asking for work has much, much better results than just waiting for something to happen. Not only does it show you as proactive, it also demonstrates an interest in the company and puts you in the forefront of your boss' mind. In addition, you have the ability to potentially tailor the work you might receive based on how/what you ask ("Is there work that needs to be done using platform XYZ? If not, I'm open to any projects you may have available.").
I definitely would not tell your boss that you're bored. At least not in those words. You can mention that you're hoping to get your hands on some more challenging projects or that you want to know if there's anything in the works, but don't tell them you're bored.
Just a side-note, here. If you really feel like you've stagnated at this job and you've confirmed that there's just no new work coming down the line for you, I'd take a moment to reconsider quitting. While there are jobs out there (especially for DBAs and programmers) that result in you having a full workload every day, many software/office jobs I've experienced have a lot of downtime. When I initially started in the industry, I was also worried that I just wasn't doing anything. I took some classes, did some practice stuff. Eventually I reached out to some friends (including a manager at the company I worked at) and they all indicated the same thing: Sometimes there just isn't any work to do. That's a product of how we view working/a work-day and not indicative of your skills as a professional.