The 24 essential etiquettes of supplication (Dua) and the means for its acceptance.

1. Sincerity to Allah.
2. To start the supplication by praising Allah and expressing gratitude to Him, thereafter sending salutations upon the Messenger of Allah (sallallahu alayhi wasallam). The supplication should also conclude in the same manner.
3. Determination in the supplication and being certain of a response.
4. The supplication should be heartfelt.
5. One should supplicate by means of using and praising the Beautiful Names and lofty attributes of Allah.
6. The supplicant may also petition Allah by means of his/her own good deeds.
7. An insistent supplication which is offered without haste.
8. To supplicate during both times of hardship and times of ease.
9. To ask Allah alone without asking by means of others.
10. To refrain from supplicating against the family, wealth, children and yourself.
11. To lower one's voice when supplicating to a moderate tone which is neither loud (so that you can be overheard) nor too quiet.
12. To confess one's sins and ask forgiveness for them and to recognise the bounty of Allah and express gratitude for it.
13. To refrain from the use of rhyme (or poetry) when supplicating.
14. To be humble and submissive and express fear of Allah.
15. To reject injustice along with your repentance.
16. Repeating specific supplications thrice and to use the supplications of the Holy Prophet (sallallahu alayhi wasallam) which are reported as Sahih.
17. To face the Qiblah (direction of the Ka'ba) when supplicating.
18. To raise the hands when supplicating but do not look upwards at the Heavens.
19. To be in the state of ritual purity (Wudhu) prior to starting the supplication.
20. Not to exaggerate in the supplications.
21. To begin by supplicating for yourself if the supplication is on behalf of another person.
22. The supplicant's food and drink, clothing and income should be from Halaal means. They should not be involved in some type of fraud or the consumption or payment of Riba (interest) on their income for example.
23. The supplicant should not ask for the severing of family ties or ask for a sinful deed.
24. The supplicant should command good and forbid evil acts and try his/her utmost to safeguard themselves of all evil deeds and sins.

References -

Regarding not lifting the gaze to the sky -
It was narrated from Abu Hurairah (radhi Allahu anhu) that the Messenger of Allah (sallallahu alayhi wasallam) said: "People should certainly stop lifting their gaze to the sky when they supplicate during the prayer, or they will lose their eyesight."
Classed as Sahih from Sunan an Nas'ai Vol. 2, Book 13, Hadith 1277

Regarding not supplicating for evil or the severing of blood ties - 

Jabir (radhi Allahu ahnu) narrated that the Messenger of Allah (sallallahu alayhi wasallam) said: “There is none who utters a supplication, except that Allah gives him what he asked, or prevents evil from him that is equal to it – as long as he does not supplicate for something evil, or the cutting of ties of the womb.” Classed as Sahih from Jami` at-Tirmidhi Vol. 6, Book 45, Hadith 3381

Regarding petitioning Allah by virtue of your own good deeds - 
It was narrated by 'Umar (radhi Allahu anhu) that; "I was a young unmarried man during the lifetime of the Prophet (sallallahu alayhi wasallam). I used to sleep in the mosque. Anyone who had a dream, would narrate it to the Prophet (sallallahu alayhi wasallam). I said, "O Allah! If there is any good for me with You, then show me a dream so that Allah's Apostle may interpret it for me." So I slept and saw (in a dream) two angels came to me and took me along with them, and they met another angel who said to me, "Don't be afraid, you are a good man." They took me towards the Fire, and behold, it was built inside like a well, and therein I saw people some of whom I recognized, and then the angels took me to the right side. In the morning, I mentioned that dream to Hafsa. Hafsa told me that she had mentioned it to the Prophet (sallallahu alayhi wasallam) and he said, "'Abdullah is a righteous man if he only prays more at night." (Az-Zuhri said, "After that, 'Abdullah used to pray more at night.")
Sahih al Bukhari Volume 9, Book 87, Number 156

Supplications & Treatment with Ruqyah from the Qur'an and the Sunnah - Dr Sa'eed bin Ali Wahf Al Qahtani. Darussalam 2009 Edition.