What is the most efficient way to destroy the universe? Don't worry... I won't actually do it, just would like to know how.
 A: We are only humans. We don't know how to destroy the universe.
A: You can't. Anything you do will only have an effect that effects an area that expands (at most) at the speed of light, so it only effects at most a sphere of radius $c t$ after time $t$. Anything else would violate the no-signalling principle which appears fundamental in physics. This means that the photon that just bounced off you into space will outrun the edge of any such an attempt to destroy the universe.
A: Find antiuniverse somewhere and annihilate it with what we have here. 
Black holes does not work as they evaporate when growing.
A: The physical process that might best be described as "destroying the universe" would be vacuum decay. Our entire universe might be in a metastable state, much like a supercooled liquid, and at some point quantum or thermal fluctuations could send some region of the universe into a lower ground state. If the region that tunnels to this ground state is large enough, it will expand at the speed of light, swallowing up the rest of the universe and converting it all to the lower state. This type of vacuum decay, in theories involving gravity, was calculated in a famous paper by Coleman and de Luccia. Anyone with the relevant background knowledge should read the paper, which like much of Coleman's work is beautifully clear and comprehensible. One passage in the paper is particularly famous (though it arguably shouldn't be taken too seriously now, given some of the assumptions it involves, including that our vacuum has exactly zero cosmological constant):

"The possibility that we are living in a false vacuum has never been a cheering one to contemplate. Vacuum decay is the ultimate ecological catastrophe; in the new vacuum there are new constants of nature; after vacuum decay, not only is life as we know it impossible, so is chemistry as we know it. However, one could always draw stoic comfort from the possibility that perhaps in the course of time the new vacuum would sustain, if not life as we know it, at least some structures capable of knowing joy. This possibility has now been eliminated."— Sidney Coleman

If we live in a metastable vacuum (and, due to the observed positive cosmological constant, most physicists working on high-energy physics, cosmology, or gravity believe that we must, though there might be some unknown loopholes), then the universe will eventually decay. Luckily, the production of critical bubbles is quite rare, allowing us the time we needed to evolve and live in this metastable universe before the inevitable catastrophe. [Edited for clarification due to Marek's comment: it's only "inevitable" if our universe really is metastable, which despite strong suggestive arguments we don't definitively know.] You would have to wait around for billions of years to witness the end of the universe. If you're more ambitious (let's hope not), you could try to cause the decay by producing a region of the true vacuum, thus catalyzing the process instead of letting it happen naturally. A number of crackpots have tried to start a panic over accelerator experiments by suggesting that they can do this. However, higher-energy collisions happen in nature, and we're still here, so this is nothing to worry about. (There are more quantitative versions of this argument, of course. The risk really is effectively zero.)
The vacuum decay processes I've mentioned don't necessarily "destroy" the universe, they just radically alter it by leaving it in a completely different state. There is one type of vacuum decay, discovered by Witten and known as the bubble of nothing instability, which is more radical and leaves behind nothing, not even a new vacuum state.
A: Kill all the turtles on which it rests.
A: I think that only an Universe can destroy an Universe (there's a lot of energy implied in doing so). If there is enough mass in our Universe, maybe its end could be a Big Crunch.
A: At CERN's website, there was - for the time being - a page (which I cannot find now, someone please help?) about how the LHC could possibly "destroy the earth, the solar system or the whole universe altogether" with possible speculations.
These included creating black holes, strange matter that is more stable than the regular one, and a bunch of other ideas which I don't remember. All of this of course rather bogus, because such collisions occur naturally in cosmic radiation at sufficient frequency.
A: Inverse the forces in nature is the fastest way to destroy it.

Make "Weak force" strong
Make "Strong force" weak

First one will increase radiation and will decay matter to energy instantaneously and second one will explode atom nucleus.
A: Dear Splashhit,
the easiest way to destroy the Universe is to continue to drive cars for years. 
Whether you do it or not, the global mean temperature will increase from 15 °C to 16 °C or so in a century. When the temperature of the materials and organisms increases by as much as one Celsius degree, they will - according to the IPCC - transform into a deadly plasma that will kill all living life forms and evaporate all stars, ending with a complete cataclysm where the vacuum itself (scared by watching the horror on its most important planet) is eaten by nothingness. If you've ever experienced the staggering heat of 16 °C, you must know what terrible thing I am talking about. It's worse than hell.
Just kidding. There is obviously no threat whatsoever that would be linked to "climate change" as this meaningless tautology is often called. The IPCC is not destroying the Universe but it is surely destroying the ability of the people to think rationally about very basic things - and it is trying to destroy the world economy, too.
Cheers
LM
A: Find a logical inconsistency and use it to prove that the universe does not exist.
You could kill all observers, that would make the existence of the universe an unprovable proposition.
Alternately you can just wait for the Big Crunch.
