He, along with co-claimants Sir Elton John and his husband David Furnish, campaigner Baroness Doreen Lawrence, politician Sir Simon Hughes, and actresses Sadie Frost and Liz Hurley, all allege ANL had a practice of "clear systematic and sustained use of unlawful information gathering".
Allegations range from tapping their phones and bugging their homes to obtaining medical records by deception.
ANL strongly denies any wrongdoing.
Monday saw the opening arguments from the claimants - all represented by barrister David Sherborne - who said the publisher "knew they had skeletons in their closet," telling the court the Daily Mail and Mail on Sunday were engaged in unlawful information-gathering (UIG) over "at least two decades".
He also said ANL's denials of unlawful acts to the Leveson Inquiry in 2011 were "not true".
In written submissions, Mr Sherborne said alleged unlawful acts "range through a period from 1993 to 2011, even continuing beyond until 2018."
Written submissions from the claimants accused the publisher of having a "culture of unlawful information gathering that wrecked the lives of so many".
Analysis: His biggest case against the press - and the stakes are high
Along with stories about the seven claimants, articles about other high-profile figures, including former home secretary David Blunkett, comedian Ronnie Barker's son, actress Billie Piper and her ex-husband Chris Evans and Tony Blair's son, Euan, were also brought up in court.
Documents released by the court showed that ANL journalists had spent well in excess of £3m on private investigators in the relevant period, including thousands of pounds paid to a "talented" voice actor, known as a "blagger" (someone, often a private investigator, who obtains private, confidential information by deception).
Prince Harry left 'paranoid' and 'isolated,' court hears
Prince Harry's case is based on 14 articles, published between 2001 and 2013, including several articles about his then-girlfriend Chelsy Davy, and another about the princes' distress over photos of their late mother, Princess Diana, in her final moments, published in an Italian magazine.
Mr Sherborne said: "It is evident from the articles and the evidence of the Duke of Sussex that the targeting of him has had a profoundly distressing effect".
He said episodes of pleaded unlawful information gathering were described by the royal as "disturbing to feel that my every move, thought or feeling was being tracked and monitored just for the Mail to make money out of it", and an "intrusion [that] was terrifying" for loved ones.
Mr Sherborne went on to say Prince Harry said it created "a massive strain" on personal relationships, while invidiously "creating distrust and suspicion", and "driving me paranoid beyond belief, isolating me".
Prince Harry will give evidence in court on Thursday.
In written submissions, the royal said: "I find it deeply troubling that Associated used phrases such as 'sources', 'friends' and the like as a device to hide unlawful information-gathering."
However, in response to Harry's claims, ANL said in their written submissions: "At all material times, the Duke of Sussex's social circle was known to be a good source of leaks or disclosure of information to the media about what he got up to in his private life."
Sadie Frost describes 'violation' in written submission
Deeply personal details about actress Sadie Frost were detailed in her claim, which the court will later hear, including her diagnosis of an ectopic pregnancy and subsequent termination.
Documents released by the court included what Frost said were "verbatim quotations" from phone conversations with her then-husband Jude Law, including a "heated argument" about his then-girlfriend Sienna Miller and details over their childcare, which were then used to inform articles published by ANL.
Describing her distress in a written submission, Frost said: "The stories they wrote violated me, my friends and family, and my children who were still so small, and they made me believe that I could not trust anyone. They were always written so close to home, like it was someone who knew me really well that was talking to the journalists. That is what I suspected at the time."
Sir Elton and David Furnish 'outraged' by alleged theft
In written submissions, Sir Elton and Mr Furnish - whose case includes articles with details about their sons and Sir Elton's health - said they felt their home and the safety of their children had been "violated" by alleged unlawful news gathering.
The court was told the couple, who followed Monday's proceedings on a live video-link, also felt "outrage" at the alleged "stealing" of their son Zachary's birth certificate and medical details after he was born.
Baroness Lawrence - the mother of murdered teenager Stephen Lawrence - said she felt like a "victim all over again", adding, "I remember feeling like I could not even grieve the injustice of what was happening privately, even for a day, or to prepare my children for the news before the world found out."
ANL say the information was gathered in legal ways, including journalists' contacts, spokespeople, freelance journalists, photographers, news agencies and prior newspaper and magazine articles.
Antony White KC, for ANL, also told the court it was "a striking feature of the case that none of the articles were the subject of complaint by the claimants at the time of publication".
The case was brought in 2022, and the following year, ANL failed to have the case thrown out on the basis that it was "time-barred" or brought too late.
It is Prince Harry's third battle with the UK press, following his win against Mirror Group Newspapers in 2023. He settled a claim against News Group Newspapers, the publisher of the Sun and the now-defunct News Of The World, last January.
The trial, which is expected to last nine weeks, continues.
