One thing that can really help you out in the beginning is to remove the top cover plates and use your finger to block the blow reed in the hole you're trying to overblow.
(I know most of my tips can be found in the link by user9328, but I think it's in the wrong order, and that the finger trick deserves more attention.)
Overblows are produce by getting the air you blow to activate the draw reed instead of the blow reed, so forcibly blocking the blow reed gets you half the way there. Overblows should NOT require very much force, especially with this technique, so just experiment with shaping your mouth cavity in various ways, sort of resembling what you do while bending, until you get it. It can be helpful to try to, very carefully and slowly, shape your mouth as is you're saying oyoyo, eyeyey or echi-echi-echi; the overblows will hopefully pop out at the consonants if you get them right. (I should probably mention that I play using a pucker technique rather than a tongue-block one, and my advice assume you do the same. It's probably similar for tongue blockers, but you'd have to ask one to know.)
After learning this, try to get this happening in an uncovered hole. On most blues harps, hole 6 or maybe 5 or 4 should be the easiest. The variability comes from your average harp not being set up for overblows, as it's considered an advanced technique, and the receptiveness is more or less random. Go through your harps, trying holes 4-6 until you find one that's a little easier. Once your skill improves you'll find more and more holes will be overblowable!
To actually get the overblow without your finger, it can be helpful to see the overblow as a two step process: first you make the blow reed choke, then you make the draw reed sound. In other words, while playing a blow note, you could first try to shape your mouth so as to make the sound go away while keeping your blow pressure. THIS step could be helped by a sudden slight increase in your blow force, especially if your just beginning, but don't overdo it. When the blow note is silenced and the air chokes, you have a split second to try and redirect the air to the draw reed. Personally, I find that mentally picture the air I blow turning downwards inside the chamber, and trying to make it so with my mouth, makes it easier to get my mouth cavity the right shape, but that might be just me. EDIT: The actual direction of the air probably doesn't matter, I just find this a useful picture to shape my mouth.
A rule of thumb while going through your harps is that very cheap harps probably won't work as well as slightly fancier ones, but a respected brand name on the covers is no guarantee overblows will work well.
You could also try to adjust the reeds to make overblows easier, but if you have a few harps I would try and see if there's any hole that's working well first. That way, you'll know what you're looking for when making your adjustments. There are a bunch of videos, just search for reed gaping and you'll be fine.