Nine people died and over 30 were injured when three no warning bombs exploded in the Co Londonderry village on 31 July 1972.
The IRA were blamed for carrying out the attacks - but never claimed responsibility - and Fr Chesney was suspected of masterminding the atrocity.
He died from cancer, aged 46, in 1980, having never been questioned by police, despite being caught with explosives in his vehicle one week after the devastating bombings.
Police at the time were reluctant to arrest the cleric for fear of inflaming the security situation in 1972. Fr Chesney, however, was questioned by the then-Bishop of Derry Neil Farren and his successor, Edward Daly, and denied any involvement in the attack.
He was transferred to a parish in Co Donegal in 1973, putting him outside the RUC's jurisdiction, following secret talks between then Secretary of State William Whitelaw and the head of the Catholic Church in Ireland, Cardinal William Conway.
In August 2010, Cardinal Seán Brady told UTV that he did not think the Catholic Church was" involved in a cover-up."
"It received this request from the highest civil authority here to help in a very difficult situation", he said.
"He (Cardinal Conway) had him questioned, not once but twice, which is more than the police did, and brought back his findings to the Secretary of State and that is where the investigation ceases."
His remarks were made on the same day that a Police Ombudsman report found that the RUC investigation into Fr Chesney was "compromised" after senior officers conspired with the Government and Church to protect him.
Ombudsman Al Hutchinson revealed that the police had intelligence suggesting that Fr Chesney continued to be involved in the IRA after the Claudy bombing.
No-one was ever brought to justice for the triple bombing. Others suspected of involvement in the attack are believed to be still alive.
So far, two families are behind the legal action, but more could follow.
They are not only suing for damages, but also for full disclosure of documents which could shed further light on Claudy.
Gordon Miller, whose father was killed in the third bomb, issued a statement on Monday evening.
The bombing of Claudy was well orchestrated, not by a few, but by many, and there are those who hold information to this day regarding those involved in that cowardly act.
"The bombing of Claudy involved issues concerning the Irish and British Governments as well as other agencies that had information regarding the terrorists involved in that terrible atrocity," Mr Miller said.
"I feel the only option left is to go down this road in an effort to root out these evil people and to hold accountable those Statutory Agencies who hold any information on the terrorists who were involved in any way with the devastation caused in Claudy."
David Temple's 16-year-old brother William was killed while helping a milkman on his round in the village.
He too issued a statement, saying: "As a family over the 41 years we have never received updates as to how the terrorists were being pursued within the police investigation."
Both families appealed for anyone with information to come forward.
Solicitor Kevin Winters is representing those launching the legal bid.
"Families are completely traumatised as a result of the ongoing failings by the State generally to investigate this case.
"But, more particularly, by recent decisions made by the police culminating in a meeting last Friday when police indicated that they were no longer any investigating leads and for that reason they were proposing to suspend the investigation.
"Now it's against that background that you have a serious deficit of information, [an] accountability deficit, and into that gap, people are left wondering, 'Well, how do we progress these serious failings?'"
He continued: "And one remedy, potentially, is to issue proceedings and to that end, we've put both the Northern Ireland Office and the Chief Constable's Office on notice, together with the - [which is] very unusual in a case like this - Catholic Church.
"We put them on notice that we intend to issue High Court civil proceedings for damages arising out of the alleged collusion which the Ombudsman found in an investigation back in August 2010."
Kathryn Eakin, 9, Protestant. The youngest of the victims, she died in the blast from the first bomb on Main Street. She had been cleaning the windows of the local grocery shop which was owned by her family.
Patrick Connolly, 15, Catholic. The teenager was caught up in the first explosion outside McElhinney's pub and shop. He died in hospital just over a week later.
William Temple, 16, Protestant. From nearby Donemana, the boy worked as a milkman's helper and was killed while on his round in the village.
Arthur Hone, 38, Catholic. A married father-of-two and an insurance salesman, he died a fortnight after the bomb attack. Two of his uncles - both priests - conducted a requiem mass at his funeral.
Joseph McCloskey, 39, Catholic. A factory worker, he died instantly when the first bomb went off.
Rose McLaughlin, 52, Catholic. A mother- of-eight, the café owner died in hospital four days after the outrage.
Elizabeth McElhinney, 59, Catholic. The owner of the pub and shop where the first bomb was planted, she was serving petrol from the shop's pump when she was killed.
David Miller, 60, Protestant. A street cleaner, he was killed by the third and final bomb contained in a mini-van.
James McClelland, 65, Protestant. A street cleaner and the oldest of the victims, he died in the third explosion.