Standing in for vika until she comes with the definitive word; it is spelled CH-A-i-K-O-V-S-K-I-i, where the lower case i represents a shortened version if the letter we transliterate as I. As to pronunciation, I will only say that I have heard it pronounced Chee kawv skee by native Russians in conjunction with the Tschaikovsky Competition. Your pronunciation sounds to me like the standard Gringo pronunciation, so I wouldn't be one to correct it.