In a large bowl, beat the cream cheese with a hand or stand mixer, until fluffy.
Add lemon pudding mix, lemon juice, and vanilla. Mix until all ingredients are combined well.
Pour in milk slowly with mixer on low/medium, until smooth and no lumps.
Fold cool whip into pudding mixture.
Put about half cup of mixture into the bottom of a 9x13 pan (this helps hold the crackers in place.) Lay down one layer of graham crackers, breaking into smaller pieces if necessary to get them to fit evenly. Spread 1/3 of the pudding mixture over the graham crackers, and repeat again, graham crackers, and 1/3 pudding mix, graham crackers, then the rest of the pudding mix (ending with pudding mix). Sprinkle crushed graham crackers on top, cover, and refrigerate overnight.