First, decide which contacts are worthy enough for special alerts. Consider prioritizing those who'd make you drop everything if they called or texted, even in the most inappropriate situations: particular friends, parents, your spouse, bosses... Beyonce.
Next, find their name in your contacts and tap "Edit" in the upper-righthand corner. Scroll down and you'll see "Ringtone" (where you can select a custom ringtone for them if you want) and then tap "Vibration." It will probably be set to the default setting.
You can select one of the standard vibration options, but if none of those floats your boat, scroll down and tap "Create New Vibration." This launches the page where you'll compose your custom lil' ditty by tapping your finger on the screen like it's a mini drum pad. You can retry as many times as you want, but once you're satisfied, hit "Save" and give it a name. That's it: you've successfully assigned your first custom vibration.