The charity has also opened an office in Moscow - the first time it has opened a base in a country against which it has a campaign.
The campaign is launched on the back of a new report on human rights abuses in Russia after the organisation denounced the Kremlin's handling of the hostage theatre drama as "yet another outrageous human rights violation".
Amnesty International's 102-page report, The Russian Federation: Denial of Justice, was launched in Moscow and claims to expose "the broad extent of human rights violations in Russia".
Amnesty International's secretary general Irene Khan said: "The awful hostage taking in the Moscow theatre is a terrifying reminder of the unsolved situation in Chechnya. We have condemned this criminal act as a despicable abuse of human rights."
The report details torture in police stations, cruel and degrading conditions in crowded prisons, racist violence against ethnic and religious minorities and torture, including rape in Chechnya.
The campaign comes four years after the charity launched a similar drive against the US. Amnesty International will lobby MPs with Russian interests, senior Foreign Office officials, Downing Street and the UK's Russian Embassy.
In another first, Amnesty International is attempting to work with local authorities that have links with Russian cities through schemes such as twinning, along with the European Union and Council of Europe.
The organisation also hopes to work with Lord Judd, who is on the Parliamentary Assembly looking at Chechnya, in a bid to raise awareness of the situation in the country.