Since viruses aren't technically "alive" in the first place, so we can't really "kill" something that isn't alive. Antiviral drugs (like Tamiflu, that was stockpiled during the swine flu outbreak 5 years ago) don't actually destroy the target but rather inhibit it from acting. Many new drugs are being developed to attack the vial reproduction cycle and prevent it, but these drugs don't actively destroy the already-existing viruses. Our cells do destroy viruses with proteins designed for breaking up threatening targets in the body. That is what antiviral vaccines do: they provide a weak or inactivated dose of the virus, so the immune system can recognize the virus and know how to destroy it if the body ever does encounter it. For rapidly mutating viruses such as the flu, the "target" given to your immune system changes every year, which is why you need annual flu vaccination. Some viruses that mutate even faster, like HIV, can't be vaccinated against yet.

Outside of the body, you can destroy viruses with a variety of methods. Alcohol (at very high concentrations only) is very effective at destroying them, for example. In labs, you can also use fire, very high temperatures, or harsh chemical treatments to destroy viruses.

Yes. The problem is in finding a way to kill the virus particles that does not harm the host more than the virus does. For example, if a person has a virus, then cremating the person would kill every viral particle in his or her body - but it would kill the person as well, so this isn't something we want to do, medically.