Yes you'll need a Russian visa for this trip. As you'll be travelling to the North Caucuses the Russian visa will also take 2 weeks to be issued rather than the standard 3-5 days. There is no express service.
The British Foreign Office do advise against travel to Chechnya. We have taken the security situation in to account with our plans for the tour.
There are two well serviced airports in the region, Grozny airport and Vladikavkaz airport. For maximum convenience we would advise flying in to Grozny and our of Vladikazkaz. Both airports are within a 2 hour taxi of each other so it is possible to join and end the tour from either airport.
As a muslim republic alcohol is quite sparse in Chechnya and bringing alcohol in with you can cause problems at the airport or land border. The restaurants of the hotel we will be using in Grozny have some of the few liquor licenses in the republic.