First possibility is to have the Cubs do better in the late 40s and first half of the 50s acquiring star players from the Negro Leagues. A Larry Doby here, a Don Newcombe there, keeping Ernie Banks per OTL would have done wonders for the talent levels of what were quite attrocious Cubs teams in this time frame.
Second possibility involves first NOT trading Lou Brock to the Cardinals. With the Cubs, Brock would have played center (pushed out with the Cards because of the fabulous glove/range of Curt Flood) and Billy Williams still plays Left. Then wait for 1970. The Cubs finished second in the NL East with an 84 and 78 record to the Pirates who finished with an 89 and 73 record. The Cubs that year were an AMAZINGLY BAD LUCK team. Projecting their Pythagorean Win Loss record (equation based on the number of runs they scored and number of runs they allowed), the Cubs SHOULD have won 94 games and only lost 68. So with Lou Brock playing center everyday and leading off with a .304 average, .351 on base percentage, and 51 stolen bases instead of an attrocious platoon of Joe Pepitone and Cleo James, the Cubs win the Division.
Once in the playoffs and anything can happen in a short series, the Cubs have a shot at beating the Reds to win the Pennant and then the Orioles or Twins to win the series.
Realistically without too many PODs 1970 is the earliest shot the Cubs have to win after 1945.
They had a pretty good team in 1972 too, so that year is another possibility.