How can a Muslim ask Allah through the Holy Quran, and is that by supplicating after reciting certain verses, or by asking Allah with a specific Surah or specific verses?

Supplicating after completing the Quran is recommended. Among the manners of supplication are to begin it by praising Allah and extolling Him, and by sending prayers upon the Prophet (peace and blessings be upon him), and to conclude it likewise. It is permissible for a Muslim to supplicate in any manner and to seek intercession through the entire Quran or part of it, such as saying: "O Allah, I ask You by the Quran" or "by such-and-such Surah"; because the Quran is the speech of Allah and one of His attributes, and it holds a great status with Him. Seeking intercession through the Quran is permissible because it is seeking intercession through one of Allah's attributes, and Allah has commanded seeking intercession through His names and attributes. It has been reported that the Prophet (peace and blessings be upon him) used to supplicate upon completing the Quran, and that the predecessors used to supplicate with the Quran.