When using a "dilute hydrochloric acid" solution, how diluted should it be?
The exact concentration doesn't matter that much. The stock solution that I used to use was 1 M aqueous HCl (~3%). If your compound is acid-sensitive then you might want to try using even more dilute HCl (e.g. 0.01–0.1 M), or skip this step altogether and just purify it using a different method. The important thing is to make sure that you have enough HCl to protonate all the basic stuff you want to remove – a relatively simple stoichiometry exercise.

I would suggest not using concentrated HCl (12 M, or ~37%) as for a typical acid-base extraction it's simply not necessary. You should save that for other circumstances, such as when you want to adjust the pH of a solution without diluting it too much with water.