I remember several years ago, bird flu was on the rise and the authorities in my country started collecting free fed chickens to prevent any infections. Somehow that changed nature's balance and we started seeing ticks everywhere. Few months later, we heard about something called the Crimean Congo disease which is transferred through ticks. Everyone was so worried about the ticks and the media did a lot of news showing us how to properly remove and get rid of ticks.
We were told to first suffocate the tick by placing a cotton ball dipped in oil on top of it. The oil prevents the tick from breathing and it also prevents the tick from releasing poisonous substances into the bloodstream as a reaction to being removed. After doing this for about a minute or so, they said that it is safe to pull the tick out with tweezers. I think the best way to do this is to pull it straight out, not twisting or turning as that might leave the head of the tick inside.
The health authorities actually recommended us not to destroy the ticks during that time period. They asked people to put the tick in a jar or other closed container and take it to the hospital with them for testing. They wanted to know if the ticks carried any disease.
I guess if there is no such risk or message from the authorities, you can kill it. We had to remove and kill a tick on my dog once and after doing the steps I mentioned, we burned the tick by holding it to fire with the tweezers. That is a very hygienic way to get rid of it as well.