No, sorry. Alice Cooper is a Shock Rock group that started during the 1970's, and it still going today. It has always been lead by front man, Vincent Damon Furnier, who is better known as Alice Cooper (also).
Alice In Chains is a completely different band, who just happens to have the name Alice in their band name. Many people have been lead to believe by this, that they have some sort of connection with the Shock Rock category creator, Alice Cooper.
There is a large difference between the bands music too, as the Alice Cooper band started the category Shock Rock with their shocking musical and theatrical style.
Whereas, Alice In Chains is most widely recognized as a Grunge Band. :)