Although Chechen had existed as a distinctive language for thousands of years, the first known Chechen language script only appeared in the 18th Century and used Arabic letters.
It wasn’t until 1862 that the first Cyrillic-based alphabet also emerged, written by the Chechens Qedi Dosov and Uslar. The emergence of the Cyrillic alphabet coincided with the defeat of the Caucasian Imamate and was part of the policy of Russification as Chechnya was brought under the Russian yoke.
In 1923 another Chechen alphabet was introduced using the Latin script, which found wide adoption in Chechen schools.
As Soviet authorities became more hardline in their attitudes however, they mandated that only the Cyrillic alphabet be used to write in Chechen (Arabic never fully caught on for various reasons). The logical culmination of this was the eventual relegation of the Chechen language itself to the status of a ‘foreign language’ in schools, taught as its own subject with everything else taught in Russian.
The fact that Russian language was a requirement for every job further entrenched as the primary language in Chechnya, leading to a total impoverishment of the Chechen language.
In short, the Chechen language uses the Russian alphabet today because it is a way of extending Russian cultural and political power into the region.