How to do the Koan study?

All of the answers given so far are correct, but ultimately incomplete. Your practice will not be effective until you learn to work with samadhi. Koans will never reveal themselves to you unless you sit facing emptiness. There's a stock passage from the sutras that describes the nuts and bolt of koan study (and really all forms of insight practice):

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to..." (MN 4)

We enter samadhi to brighten our minds and temporarily rid ourselves of our coarser defilements. When we shift these obstacles to the side, we are better able to work with our original minds and directly see what the koan is pointing at. Our smaller mind is simply unequipped to perceive what the bigger mind is capable of investigating. It works with logic and language. The big mind we access through samadhi, on the other hand, works with the unintelligible fragments of our store house consciousness that koans seek to address. Put another way, trying to answer a koan without samadhi is like trying to hear a symphony with your nose. Your nose is great if you want to smell a flower, but it is useless for the purposes of listening. Koan study requires that we direct the proper, purified sense to the case we are given.

Before you embark on koan study, it is absolutely essential that you learn to purify your mind and body. Work with your breath. Master doing nothing. Mingle your eyebrows with those of the patriarchs and pass the barrier where the foaming billows wash the sky. Until you do, koans will remain inaccessible.

"Hey, Newbie! What was your original face before you were born?"

Choose ONE koan. Not two. Of the two you refer to, I recommend 'original face', unless you are spending a LOT of time with your teacher, as the second one will become a philosophical fiasco otherwise.

Don't rationalise. It's not a philosophical or intellectual exercise. Ask yourself the question when you are not thinking about it.

Lots of Zen practitioners write it down. Paint it. Draw it, etc. Does it help? Maybe. Maybe not.

Don't talk about it. It's your practice. Don't talk about your practice except to your teacher. Maybe your students, one day.

Importantly, it's not a question about rebirth. You don't need to believe in rebirth in order to realise this koan.

Also, and paraphrasing @OyaMist's answer, also, there's nothing special about you either. So when you perceive anything, you can ask yourself in that moment: "what was its original face before it was born?", whereas for @OyaMist, it's often "does it have the buddha nature?"

The main, core, essential process? Just sit.

It's going to kill you. That's the point. But don't lose your life. If you are on the correct path, the path itself is an enriching and empowering experience. Hard work sometimes, painful sometimes.

I have heard of students kicking and screaming; having to be literally dragged to sanzen. Maybe ask yourself: Why is that?

Maybe a koan is provided, maybe by The Teacher, when appropriate, and might be selected by The Teacher for the Student: there are a number of documented koan. Asking what to do with it/them, eg. is a start. Maybe some patience, and typically koan mightn't be suggested to someone right away etc. Describing koan with words is sort of counter to koan. Maybe consider it some, and develop quiet contemplation. Thats maybe more when addressing koan would be done more. Maybe aren't lots of specific schedules for it either in terms of months, years etc. Maybe write them down, or, they're brief enough, may be able to remember them. Attempting to consider koan at the outset & before some development of quiet contemplation may tend to result in less effectiveness.

Koans break the intellectual mind and require deeper investment.

SN41.6:1.4: “Householder, there are three processes. Physical, verbal, and mental processes.”

Practice a koan physically, verbally and mentally. For example, walk with it, speak it and feel it.

This sounds mysterious, but is actually practical. To walk with a koan, one might attend to "left foot original face", followed by "right foot original face". When meditating, one could silently speak the koan. To feel it one could look at anything that appears as "this tree original original face feels tall", "this car original face feels fast". And so forth. In other words, tie the koan into everything seamlessly.

In this way one discovers "original face" and can answer the Roshi by presenting the original face. It takes much practice because most of us are distracted by other stuff day-to-day and moment-by-moment.

My koan was Joshu's Mu. Everything was Mu. Same practice, however.