Your main goal should be elimination of as few characters as possible and doing that as painless as possible. So I recommend approaching your targets from behind, surprising them and not giving them any chance to fight back. What's interesting, the game allows you to choose whether you want to strangle your opponents or break his neck. Strangling allows you to keep your enemy alive but lasts longer so you have to make sure that you have enough time to finish the process. Breaking the neck lasts only few seconds but it is connected with another victim on your account. So if you do not care about keeping given person alive, it is strongly recommended using a garrote. This gadget has one more advantage - after killing the person with it, you do not have to pick up the corpse but you're able to drag it right away.
I strongly advise not to use melee weapons or elements of the environment, to inflict lethal blows in the head, because all such attacks are noisy and there is a big risk that you'll alert nearby enemies. You should also rarely rely on close combat, consisting on pressing on-screen keys/buttons. You could easily make a mistake here and the whole struggle might attract attention of other hostile characters in the area.
Elimination of opponents with usage of firearms is completely separate subject. First of all try to use weapons with silencers, thus reducing a risk of triggering an alarm. It is also vitally important to aim at head of your opponents, being sure that single shot will do the trick (enemies with headgear are fortunately rare).
Regardless of how you've eliminated your opponent, it is very important to get rid of the body. A big advantage is that almost every explored area has at least a couple of large crates and cabinets, which can be used in this purpose (each such object can contain two bodies at maximum). If the person was murdered, you might consider also throwing the body over the railing or dropping it in a hole.