The Sistine Chapel located on the territory of the Vatican City State, more precisely, in its worldwide famous museums. It was built by Pope Sixtus IV within the Vatican right to the north of St. Peter’s Basilica. The territory of the Vatican is not considered as Rome’s city center and it takes time to reach it. For instance, from the Colosseum to the Vatican area you can get bus №81 (from S. Gregorio bus stop to Risorgimento bus stop) and walk by foot to the Vatican Museums and Sistine Chapel.
Another option is Roman metro: first, go to the Colosseum metro station and choose the direction to Termini Station. From Termini, choose the Line A which goes to Valle Aurelia metro stop and there will be only 5-min walk to the entrance of the museums. However, you should better check google maps to find the best way to reach the Vatican museums, depending on your location.