When you say "I lack five apples", it means that you're five apples short of some requirement, e.g., there are twenty-five people coming to dinner, desert is baked apple, but you have only 20 apples. People who consistently do stupid things are said to "lack common sense".
When you say "I don't have five apples", it means that someone's asked you for five apples, e.g., on Halloween, there are five trick-or-treaters on your doorstep and you don't have an apple for each, but you don't have anything else to give them either.
Writing or saying "I do not have five apples" is really weird to me unless it's an example of how to say and write "don't" without using the contraction, which almost all native speakers use most of the time, except when they want to emphasize the negative: "No, I do not have five apples, contrary to your pigheaded belief that I do!"
"Don't" really is an official word and has been for centuries. It's used all over the anglophone world. It's discouraged or prohibited only in formal expository prose and in classrooms run by anal-retentive prescriptivists who think it's "vulgar". One of the great achievements of the early Renaissance was publishing books and newspaper in the vulgate. The King James Bible online shows 4,460 instances of "don't" in its modern translation (that's official enough for many Americans).