Find a grip that is comfortable for you, you can do palms facing out, palms facing in, hands close together, or far apart, find what works best for you.
Tilt your head back so that your chin extends into the air, since only your chin must pass the bar for the rep to be counted you can reduce the distance your need to travel by doing this
Drop down quickly into the dead hang position, the longer it takes to go return to the bottom, the more energy your wasting. Also minimize the amount of time you spend hanging, as this burns energy as well.
Don’t forget to breathe; breathe in on your way down, and then push out the air on your way up, as with any exercise proper breathing will help you a lot.
Try switching grips; you are allowed to change your grip on the bar as long as you don’t touch the ground, so if you find you cant do any more, give it a try, its a good way to add on 1 or 2 pullups right way.
The picture below gives you a good idea of what your hand should look like on the bar, the position of the thumb does make a notable difference so if your doing it like the “not recommended” give the other way a try and see if it works better for you, if it doesn’t, do whats more comfortable for you.