Is it permissible for someone to spy on someone else's personal computer to discover their secrets and e-mail messages under the pretext that he wants to advise and warn them against what they are doing?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Following up the rumours about others and spying on them are prohibited, whether or not it is done to find out their sins or out of mere curiosity. Allaah The Almighty Says (what means): {O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin. And do not spy or backbite each other. Would one of you like to eat the flesh of his brother when dead? You would detest it. And fear Allaah; indeed, Allaah is Accepting of repentance and Merciful.} [Quran 49:12]

Commenting on this verse, Ibn Hajar Al-Haytami, a Shafi‘i scholar states that the above-mentioned verse stresses that it is prohibited to search for the hidden secrets of others. It is also narrated on the authority of Abu Hurayrah that the Prophet , forbade spying on, following or listening to comments on other people's secrets. [Al-Bukhari and Muslim]

Simply surmising that someone is committing a sin or a violation of the Sharee‘ah does not permit others to spy on them, for Muslims are in principle innocent of sins and violations unless proven otherwise. It is also impermissible to spy on others even if there are indications that they are committing sins or violations, unless  it involves trespassing on the rights of others.

Al-Qaadhi Abu Ya‘la, a Hanbali scholar mentioned that spying and detection are permissible in case of major violations, such as adultery or murder. This applies as long as the case is informed by a trustworthy person, so that the situation would  not  worsen. However, spying is impermissible if the violation is less serious.

Ibn Muflih, a Hanbali scholar mentioned that Imaam Ahmad stated that if someone sees a bowl containing an intoxicant, he should not investigate it.

What you are asking about  comprises  thinking ill of Muslims, backbiting them, spying on them and blemishing their honour. All of these acts are not only  prohibited in the Quran and the Sunnah but are also reprehensible acts. That person must stop doing this, or Allaah The Almighty will disclose his faults in his own house.

It is narrated on the authority of Abu Barzah Al-’Aslami that the Prophet warned those who say they are  believers but their belief is not established in their hearts, against backbiting Muslims and spying on them. The Prophet continued to say  that whoever does so, Allaah The Almighty will disclose his sins until He scandalizes  him in his own house. [Abu Daawood]

Spying on others  under the pretext of seeking to reform and advise them are deviant thoughts inspired by Satan.  Calling to the worship of Allaah The Almighty must only be done according to the lawful methods prescribed by Allaah The Almighty. Moreover, such conduct may lead those who are being spied on to commit more sins. It is narrated on the authority of Mu'aawiyah that he heard the Prophet say that anyone who pursues others’ faults (which Allaah has concealed) has certainly corrupted them or  is on his way to corrupting them. [Abu Daawood and Ibn Hibban]

In conclusion, anyone who does so must repent and ask Allaah The Almighty for forgiveness. He should ask the betrayed persons to forgive him or take back their rights from him in this life, before they do so on the Day of Judgment when neither one's wealth nor sons will avail him - unless he comes to Allaah The Almighty with a pure heart.

Allaah Knows best.