You cannot be successful at PVP if you never do any damage. You simply cannot hurt another player unless you run up to him with a sword and slap him a few times. (Yes there is a bow and a few other ways but that's not my point.) Damage is important in PVP. Using a diamond sword versus a wooden sword is like racing a Ferrari against a Go-Cart. You will be able to kill your enemies quickly, and they will be able to do little damage.
The damage factor is a factor very much controlled by you. You can improve it by following these tips.
1. First off, equipment is the most important, with a diamond sword you can kill somebody with very few hits. Get a good weapon, iron is usually fine, before PVPing.
2. Also, to maximize damage you want to get criticals and knockbacks on your opponent. These are essential when becoming a successful PVPer. To maximize your criticals and knockbacks, sprint at your enemy and continue jumping as you whack 'em. This will knock them back and you will preform criticals on them, allowing you to kill them very quickly with minimal damage taken yourself.
3. Potions and enchantments are ideal to improving your damage factor. A strength potion can increase your damage incredibly, and an enchanted sword can have amazing capabilities. If possible, always try to enchant your items and stock up on potions.
4. Lastly, commit. If you cannot decide whether to retreat or fight on make up your mind, or you will simply die. If another player is running at you they will get a knockback on you along with a few criticals. This will allow them to do a lot of damage before you are able to hit them. You can not sprint backwards, so you will be walking while they are jumping and sprinting getting their damage factor insanely high due to Tip #2. Basically, if you are over matched run, if you think you can handle the fight, fight on!