It is mustahabb to complete the Quran in Ramadan
Question

Can you please tell me, is it nessecary for muslims to finish the whole Quran during the month of Ramadhan? If so, can you use a hadith to back this up?.
Praise be to Allah.The questioner is to be commended for his keenness to find out the ruling on this issue with evidence. Undoubtedly this is something essential, which every Muslim should strive to do, so that he will be following the Quran and Sunnah. 
Al-Shawkaani (may Allah have mercy on him) said in Irshaad al-Fuhool (450-451): 
Undoubtedly the common man should ask the scholar, and the one who is imperfect should ask the one who is perfect, so he should refer to people of knowledge who are known for their religious commitment and piety. The scholar to be consulted is the one who has knowledge of the Quran and Sunnah and who is acquainted with all the knowledge he need to enable him to understand the Quran and Sunnah, so that he will be able to give him the correct answer. The common man should ask the one who is well versed in the Book of Allah and the Sunnah of the Messenger of Allah (peace and blessings of Allah be upon him), thus he will be learning the truth from its sources, and he will learn the ruling from the proper source, and he may rest assured that he has been given a sound opinion and will not fall into error that goes against Islam. End quote. 
In the book of Ibn al-Salaah, Adab al-Mufti wa’l-Mustafti (p. 171) it says: 
Al-Sam’aani stated that there is no reason why one should ask the mufti for evidence, so as to be on the safe side. He should tell him the evidence if it is definitive, but he does not have to do that if it is not definitive, because the commoner may not have enough knowledge to understand the ijtihaad. And Allah knows best what is correct. End quote. 
Secondly: 
Yes, it is mustahabb for the Muslim to read Quran a great deal during Ramadan and to strive to complete it, but that is not obligatory, i.e., if he does not complete the Quran he is not sinning, but he has missed out on a great deal of reward. The evidence for that is the report narrated by al-Bukhari (4614) from Abu Hurayrah (may Allah be pleased with him) who said: Jibreel used to review the Quran with the Prophet (peace and blessings of Allah be upon him) once every year, and he reviewed it with him twice in the year in which he passed away. 
Ibn al-Atheer said in al-Jaami’ fi Ghareeb al-Hadeeth (4/64): 
i.e., he used to study with him all that had been revealed of the Quran. End quote. 
It was the practice of the salaf (may Allah be pleased with them) to strive to complete the Quran in Ramadan, following the example of the Prophet (peace and blessings of Allah be upon him). 
It was narrated that Ibraaheem al-Nakha’i said: al-Aswad used to complete the Quran in Ramadan every two nights. Al-Siyar (4/51). 
Qataadah used to complete the Quran in seven days, and when Ramadan came, he would complete it every three days. When the last ten days came, he would complete it every night. Al-Siyar (5/276). 
It was narrated from Mujaahid that he used to complete the Quran every night in Ramadan. Al-Tibyaan by al-Nawawi (p. 74). He said: Its isnaad is saheeh. 
It was narrated that Mujaahid said: ‘Ali al-Azdi used to complete the Quran every night in Ramadan. Tahdheeb al-Kamaal (2/983). 
Al-Rabee’ ibn Sulaymaan said: al-Shaafa’i used to complete the Quran sixty times in Ramadan. Al-Siyar (10/36). 
Al-Qaasim ibn al-Haafiz ibn ‘Asaakir said: My father used to pray in congregation and read Quran regularly. He would complete it every week, and every day in Ramadan. Al-Siyar (20/562). 
Al-Nawawi (may Allah have mercy on him) said, commenting on how often the Quran should be completed: 
The best view is that that varies from one person to another. The one who is seeking to understand it and ponder its meaning should limit himself to as much as he can understand fully when he reads, and the one who is busy spreading knowledge or other religious works, or working for the public interests of the Muslims, should limit himself to what will not cause him to neglect his work. 
If he is not among the categories mentioned here, then he should do as much as he can without reaching the point of boredom. End quote.
Al-Tibyaan (p. 76) 
However it is mustahabb to read Quran and complete it in Ramadan, and that remains mustahabb but it is not one of the obligatory duties and the Muslim is not sinning if he does not do it. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: Is it obligatory for the fasting person to complete the Quran in Ramadan? 
He replied: 
Completing the Quran in Ramadan is not obligatory for the fasting person, but he should read the Quran a great deal in Ramadan, as that is the Sunnah of the Messenger of Allah (peace and blessings of Allah be upon him), and he (peace and blessings of Allah be upon him) used to review it with Jibreel every Ramadan. End quote. 
Majmoo’ Fataawa Ibn ‘Uthaymeen (20/516) 
See also questions no. 66063 and 26327 
And Allah knows best.