Generally speaking when a recipe calls for a cup of a fresh herb like cilantro, how tight to I have to cram it into the measuring cup?
Oh man. That's a lot of cilantro. Some people are more sensitive than others, but make sure you like it a lot before putting that much in a dish.
In a perfect world we would all have ready access to scales and all recipes would list ingredients like these by weight instead of volume. That said, reality tends to lean more in favor of the volume-based approach.
How much you pack an ingredient varies by ingredient-to-ingredient. Brown sugar is a great example of an ingredient that is traditionally packed firmly into the measuring cup.
For an herb I would say "almost none" is the right amount of packing. Recipes will usually call for a minced or chopped volume. The herbs should settle down into the measuring cup fairly well. Fill it to be level with the brim like you would with other ingredients. Also, be sure to check if the recipe calls for "1 cup chopped cilantro" or "1 cup cilantro, chopped" as those could indicate if the measurement is to be made before or after chopping.
The best part about herbs is that it's completely subjective. Cook to your preference and make a note for next time. Adjust over time to find the right answer for you. As Aronut says in this answer, "Volume measurements of herbs are hopelessly imprecise to begin with... The most reliable way to know if you've got the right amount is to simply taste it."