There is a relationship between aspartame and cancer, but the quantity needed to consume for these risks to be fulfilled, is much more than than what almost anyone drinks. According to both the World Health Organization and the FDA, " the acceptable daily intake of aspartame at 50 milligrams per kilogram of body weight per day and says approved sweeteners are safe when used in moderation" (Karimi). This translates to " 17-18 cans of diet soda, 75 Equals or Nutrasweets, 23 Splendas, Sweet Ones or Newtames, 45 Sweet’N Lows, 9 Truvias" which is significantly more than what most people consume in a week. The American Cancer Society has come to a similar conclusion, stating, "The assessments of aspartame have indicated that, while safety is not a major concern at the doses which are commonly used, potential effects have been described that need to be investigated" (ACS). Ultimately, consuming diet coke or aspartame is completely safe, unless, consumption reaches huge amounts, and even then, the risk of developing anything increases, but is not guaranteed.
https://www.cityofhope.org/aspartame-and-cancer-risk-should-you-put-down-the-diet-coke
https://www.cancer.org/cancer/risk-prevention/chemicals/aspartame.html