How Did the Prophet Clean Himself after Relieving Himself?
Question

I would like to know how the Prophet (blessings and peace of Allah be upon him) cleaned himself with water (Istinja’) after relieving himself, especially after urinating. Are there any reports in the Sunnah about that? Did he pour water on himself, or did he take water in his hand and wipe? What is the correct way? And how much water should be used to do as the Prophet (blessings and peace of Allah be upon him) did? And how did he (blessings and peace of Allah be upon him) purify his hand afterwards?
Praise be to Allah.The prohibition of extravagance in using water 
The Messenger of Allah (blessings and peace of Allah be upon him) forbade extravagance in all things, and he was economical in his use of water when purifying himself; he would not use too much of it, to the extent that he would do Wudu with a Mudd of water , as Anas (may Allah be pleased with him) said. (Narrated by Muslim (325)
A Mudd is the amount of water that can be held in the two cupped hands of a man of average build.
Similarly, he (blessings and peace of Allah be upon him) was not extravagant in his use of water when he cleaned himself after relieving himself (Istinja’); he did not use any more than necessary, which is the amount needed to remove the impurities from the place.
How did the Prophet clean himself after relieving himself?
It was the practice of the Prophet (blessings and peace of Allah be upon him), if he needed to clean himself after relieving himself , or to remove any filth, to do that directly with his left hand.
Abu Dawud (33) narrated that `Aishah (may Allah be pleased with her) said: The right hand of the Messenger of Allah (blessings and peace of Allah be upon him) was for his Wudu and food, and his left hand was for cleaning himself after relieving himself and removing any filth. (Classed as authentic by Al-Albani in Sahih Abu Dawud)
Al-Bukhari (265) and Muslim (317) narrated that Ibn `Abbas (may Allah be pleased with him) said: “Maymunah (may Allah be pleased with her) said: I brought water to the Messenger of Allah (blessings and peace of Allah be upon him) to do Ghusl (ritual bath) with it. He poured water on his hands and washed each one two or three times, then he scooped water with his right hand onto his left hand and he washed his private parts. Then he rubbed his hand on the ground – according to one report: then he rubbed his hand on the ground, or on the wall. – 
So if he was able to scoop water from the vessel with his right hand and wash with his left hand, he did that.”
Al-Bukhari (150) and Muslim (271) narrated that Anas ibn Malik (may Allah be pleased with him) said: “When the Prophet (blessings and peace of Allah be upon him) went out to relieve himself, another boy and myself would bring a vessel of water, meaning for him to clean himself with it.”
The word translated here as vessel refers to a small leather vessel that is used to carry water, as is stated in Fath Al-Bari by Ibn Hajar (1/76).
Ibn Rajab (may Allah have mercy on him) said:
“If the water was in a small leather vessel and the like, he would pour the water from it onto his private part." (Fath Al-Bari by Ibn Rajab  1/276).
The Prophet’s practice after leaving the place where he relieved himself
When the Prophet (blessings and peace of Allah be upon him) came out of the place where he went to relieve himself, he would rub his hand on the ground, as noted above in the Hadith of Maymunah (may Allah be pleased with her), then he would “wash his private parts. Then he rubbed his hand on the ground.”
An-Nasa’i (50) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) did Wudu, and when he cleaned himself after relieving himself, he rubbed his hand on the ground. (Classed as sound by Al-Albani in Sahih An-Nasa’i.)
This action is clear and the meaning is understandable; it was in order to remove anything that may have got stuck on the hand as a result of cleaning oneself, such as filth or an unpleasant odour. Hence Imam Al-Bukhari (may Allah have mercy on him) included it in a chapter entitled: “Wiping the hand with dust to clean it.”
It says in `Awn Al-Ma`bud (1/44):
“In order to remove an unpleasant odour, if it remains after washing the hand.”
If the individual washes his hands with soap and the like, which will remove such traces, this is similar to rubbing it on the ground, or maybe better.
An-Nawawi (may Allah have mercy on him) said:
“It is recommended for the one who has cleaned himself with water, after he has finished, to clean his hand with dust or potash, or to rub it with dust or on the wall, to remove any off-putting traces from it.”(Sharh Muslim, 3/231)
Then he (blessings and peace of Allah be upon him) would do Wudu as for prayer; he would start by washing his hands three times , before putting them into the vessel.
Al-Bukhari (265) and Muslim (317) narrated that Ibn `Abbas (may Allah be pleased with him) said: “My maternal aunt Maymoonah told me: I brought the Messenger of Allah (blessings and peace of Allah be upon him) water to do Ghusl from Janabah (a state of ritual impurity due to semen discharge). He (blessings and peace of Allah be upon him) washed his hands two or three times, then he put his hand in the vessel and poured some water over his private part and washed it with his left hand. Then he struck his left hand on the ground and rubbed it vigorously. Then he (blessings and peace of Allah be upon him) did Wudu as one would do before praying, then he poured three handfuls of water over his head, then he washed the rest of his body. Then he  (blessings and peace of Allah be upon him) moved away from the spot where he had been standing and washed his feet.”
According to a report narrated by Muslim: “He washed his hands two or three times, then he put his hand in the vessel.”
And Allah knows best.