How should missed prayers be made up?

Firstly: 

It is not permissible for a Muslim to delay the prayer until its time is over with no excuse. Allah says (interpretation of the meaning): 

“Verily, As‑Salaah (the prayer) is enjoined on the believers at fixed hours” [al-Nisa’ 4:103]

i.e., at specific times. 

Reasons why prayers may be delayed until their time is over include sleeping and forgetting. It was narrated that Anas ibn Maalik said: The Prophet of Allah (peace and blessings of Allah be upon him) said: “Whoever forgets a prayer or sleeps and misses it, its expiation is to make it up as soon as he remembers it.” Narrated by Muslim, 684. 

With regard to work, study, etc, these are not excuses that allow delaying a prayer until its time is over. Allah praises some people by saying (interpretation of the meaning): 

“Men whom neither trade nor sale (business) diverts from the remembrance of Allah (with heart and tongue) nor from performing As‑Salaah (Iqaamat‑as‑Salaah) nor from giving the Zakaah. They fear a Day when hearts and eyes will be overturned (out of the horror of the torment of the Day of Resurrection)” [al-Noor 24:37]

Secondly: 

Whoever neglects to pray until the time for that prayer is over, with no excuse, has committed a sin which is one of the major sins. He has to repent to Allah and resolve to offer that prayer regularly on time. Making it up after its time is over will not avail him anything when he has missed it with no excuse. He should also do a lot of naafil (supererogatory) prayers, in the hope that they will make up the shortfall in his obligatory prayers. 

With regard to the one who delays a prayer until its time is over because of a (legitimate) excuse, such as sleeping or forgetting, he has to perform the prayer as soon as that excuse is no longer in effect, because the Prophet (peace and blessings of Allah be upon him) said: “Whoever forgets a prayer let him offer it as soon as he remembers it , and there is no other expiation but that.” Narrated by Muslim. 

He should pray the number of rak’ahs he would have prayed if he had offered it on time, no more and no less, and without changing the way it is done. 

In the hadeeth of Abu Qataadah in Saheeh Muslim (681) there is the story of how the Prophet (peace and blessings of Allah be upon him) and his companions slept and missed Fajr prayer when on a journey, and (they did not wake up) until the sun had risen. Abu Qataadah said: “Then Bilaal gave the call to prayer, and the Messenger of Allah (peace and blessings of Allah be upon him) prayed two rak’ahs, then he prayed Fajr, and he did the same as he did every day.” 

Al-Nawawi said: “The same as he used to do every day” indicates that the way in which missed prayers are to be made up is the same way as they would normally be done. 

The basic principle according to the scholars is that making up is identical to the original action.  

And Allah knows best.