This is probably the easiest of the 'Win their first World Series' achievements. The Cubs have a good lineup and a good rotation to begin with so you don't need to make as many moves as you do with the other teams, which will minimize the time that you spend reloading your save game.
Step 1) Resign your players to contracts. Some players will be asking for more money than others. You can save money against your cap by lowering their asking price, increasing their years and giving them a player option for something astronomical....like $30 million.
Step 2) Analyze your needs. The Cubs have a good offense, anchored by Lee, Soriano and Ramirez, and with good complementary players in Theriot, Fontenot, Pie, Soto. Their pitching starts with Zambrano and has decent 4th starters in Marquis and Lilly, but needs some help to fill in the gaps. After players are not offered arbitration on November 14 you'll see some bigger names available in the free agent pool.
Using the same trick mentioned above (lowering base salary while offering massive player option) you can probably pick up 2 or 3 free agents. I signed Sabathia to about $15 M a year and Lackey to a deal also about $5 million less than his asking price. Derek Lowe would be a decent third option. Note that I've never seen Rich Harden perform well in this game so don't count on him.
Step 3) Since the Cubs are pretty stacked I didn't need to go the incredibly tedious route of playing 2-3 games, saving, and reloading if things don't go well. I simmed a month at a time, saving if the Cubbies were within a couple games of the division lead. By September they were 2 out...at this point I would sim each game individually and maybe 2 or 3 in a row.
I got this achievement in a little more than an hour...which pales in comparison to how long it took me to get some of the others.