It seems to be a mixture of black magic and body transference
From The Case of Charles Dexter Ward, we see Joeseph Curwen was almost certainly a necromancer:
In many cases, diarists have recorded with some awe, Curwen shewed almost the power of a wizard in unearthing family secrets for questionable use. During the final five years of his life it seemed as though only direct talks with the long-dead could possibly have furnished some of the data which he had so glibly at his tongue’s end.
Whereas in The Thing On The Doorstep, Asenath Derby is ... just freaking evil?
“Asenath has gone, Dan. We had a long talk last night while the servants were out, and I made her promise to stop preying on me. Of course I had certain—certain occult defences I never told you about. She had to give in, but got frightfully angry."
But clearly she is an occultist of the Arkham variety. In both cases we see physical metamorphoses:
In Charles Ward, it is a bit more subtle. He's an ancestor and already looks a bit like Curwen - but...
For above the young man’s right eye was something which he had never previously noticed—a small scar or pit precisely like that in the crumbled painting of old Joseph Curwen, and perhaps attesting some hideous ritualistic inoculation to which both had submitted at a certain stage of their occult careers.
In poor Edward Derby's case, the transformation is a bit more severe:
The face beside me was twisted almost unrecognisably for a moment, while through the whole body there passed a shivering motion—as if all the bones, organs, muscles, nerves, and glands were readjusting themselves to a radically different posture, set of stresses, and general personality.
And he's not shy about saying that Asenath is controlling his body, and the technique can be used to gain immortality:
Dan, Dan, don’t you remember him—the wild eyes and the unkempt beard that never turned white? He glared at me once, and I never forgot it. Now she glares that way. And I know why! He found it in the Necronomicon—the formula. I don’t dare tell you the page yet, but when I do you can read and understand. Then you will know what has engulfed me. On, on, on, on—body to body to body—he means never to die.
So basically dark magic, mind transference, mind transference, body transformation and voila - you're a lich. There are some indications this is an ability held and given by Old Ones known as Yithians. Asenath may have even been a hybrid.
To answer your questions specifically:
How one can became a lich? I believe that there may be a ritual of some sort, but which one?
There are references to the Necronomicon and lots of indications it requires some kind of strong occultist magic. It doesn't so much seem a single ritual as a repeated assault on another's body. Curwen could apparently do this even with the slight inconvenience of being dead. Well, mostly incorporeal at least. Asenath does the same trick towards the end of her tale.
How a lich can be destroyed? It requires a ritual? Which one?
Willet defeats the now corporeal Curwen with magic:
An eye for an eye—magic for magic—let the outcome shew how well the lesson of the abyss had been learned! So in a clear voice Marinus Bicknell Willett began the second of that pair of formulae whose first had raised the writer of those minuscules—the cryptic invocation whose heading was the Dragon’s Tail, sign of the descending node—
At the very first word from Willett’s mouth the previously commenced formula of the patient stopped short. Unable to speak, the monster made wild motions with his arms until they too were arrested. When the awful name of Yog-Sothoth was uttered, the hideous change began. It was not merely a dissolution, but rather a transformation or recapitulation; and Willett shut his eyes lest he faint before the rest of the incantation could be pronounced.
Asenath is defeated through pure physical means, with Upton in prison for having killed her while she inhabited Edward's body. However, if she is doing a trick similar to Curwen's, then it could be that she could resurrect herself.
What are the benefits and the drawbacks from being a lich?
Benefits: immortality and body swapping.
Downside: You're probably going mad reading that Necronomicon thing and your neighbors have a tendency to try and murder you (and possibly burn down your house).
How a lich moves to a new body?
In Charles Ward's case it seems that Curwen can do this because they are related. This is even referenced in the opening quote:
a Philosopher may, without any criminal Necromancy, call up the Shape of any dead Ancestour from the Dust whereinto his Bodie has been incinerated.
Although it should be noted he does plenty of criminal Necromancy in the process. Asenath seems to just do it to her husband by pure magic and spite.