If you gave the other person your Dropbox user ID and password, then it should be enough to delete the account. Once the account is deleted, he won't be able to access it through the web or the desktop program. This assumes that he didn't download the files or install Dropbox on his computer. If he did, there's no way you can force the files to delete themselves.
If you actually share files from one Dropbox account to another (using Dropbox's share feature, not by sharing your account), then you have a little more control. This Dropbox Help Page says that you can remove the files completely when you remove a user from a shared folder. This still depends on him not copying the files to another place on his computer.