Sending things to outer space is crazily expensive in terms of fuel; the same holds if you aim for the Sun - the common idea that it's easy is very wrong. The Earth is moving around the Sun at 30km/sec, you need to burn off most of that velocity to actually hit it. For leaving the Solar System, the Internet provides the value of around 40km/sec. You can get most of those speeds by clever use of slingshot maneuvers though, and this is the way we usually do missions to outer planets.
Also, honestly, I don't see the point. The amount of nuclear waste we produce is minuscule compared to other equally or more dangerous chemical substances. Besides wasting perfectly good rocket fuel, destroying it this way would rob us of opportunity to reuse it in the future - hell, we already know how to reprocess some of the spent nuclear fuel for next-gen fission reactors!