As in many engineering problems, we can start by looking at how nature does it. In general, we see three common methods by which animals (and plants) poison one another. Contact poisons, injected poisons, and poisonous objects which are left in the targt.
This lovely little strawberry-looking dude is a poison dart frog. His method of poisoning is simple: touch him and die. This is true even for humans, though something like gloves would make it safe for a person to handle him.
Weapons could be coated with such a contact poison fairly easily. With something toxic enough that touching it would be lethal, slashing or stabbing someone would be lethal as well. The downside is that you have to be incredibly careful with contact poisons. Brush up against your sword? Dead. Get a little on your finger while coating your blade? Dead. Only a very cautious expert would want to use the sorts of poisons that will kill with a touch.
Many animals, rather than just letting a poison seep in through the skin, actively seek to inject poison into their victim. This has the advantage of getting the poison (or rather, the venom) right into the blood stream. This is common in predators, such as snakes and spiders, and is also used defensively by nudibranchs, though they cheat and steal the poison injectors off of jellyfish tentacles instead of making their own.
Injected venom isn't as lethal as contact poison, so it would be more manageable for a soldier to handle without killing himself. The difficulty is that it needs to be introduced into the victim's blood stream. Luckily, swords and spears are purpose-made for introducing themselves into a hapless victim's blood stream, so coating a weapon and slashing might still do the trick.
It also might be possible to develop a weapon that can do a better job of injecting its venom. You could, for example, create a spear with a hollow tip filled with pressurized venom. If the tip of the spear was a spring loaded valve that would get pushed in upon contact with the target, it could effectively pump a load of venom into any wound it made. The downside would be that all of this hollowing out of the spear head would make it more brittle, and the valved design probably wouldn't fare as well against armor. It would also need to be refilled and repressurized regularly, as the contents of its venom chamber would likely be spent after a blow or two.
That's a bee stinger, plus the tip of its abdomen containing a venom sac and some muscle to keep it pumping once the barbs on the stinger lock it into its target. Some plants use this technique as well, leaving thin hairs filled with venom in your skin when you brush up against them. Some sea urchins have similar barbed, venomous spines that break off and burrow into the skin of unlucky divers that brush up against them.
The advantage of something like this is that, once your stinger embeds itself in the target, it will keep pumping more venom into them. The venom, like our injectable venom, need not be as nasty as contact venom. For something like a melee weapon, your best bet for going down this route would probably be a spear with a detachable spear head. The spear head could either be filled with venom, or else just made of something poisonous. If it were barbed and not attached too firmly to the shaft, it could be stuck into the target and would remain when the shaft of the spear was removed.
This approach would inject more venom, without the need for a system of rapidly pumping the venom into the target. It would also leave a big hunk of metal lodged in the skin of whoever it was used on, which would further add to the plight of that unfortunate soul.
The downside is that your spear no longer has a tip. I'd recommend making a second, general purpose stabbing tip on your spear and lashing the venomous tip on top of that one if you're going to use this approach. Of course, that will add weight, so either way your spear won't be as effective as a normal one.
Why poison still might not be worth it
Ultimately, there is still one problem with poison, regardless of what kind you use and how you inject it into your target: it doesn't act particularly quickly. A fast acting incredibly deadly toxin, like that found in poison dart frogs, still takes tens of minutes to kill its victim. Other poisons could take even longer. Hunters and snakes that use poisons to catch their food generally rely on poisoning the target and then following it around until it dies.
In close combat, tens of minutes is still enough time for the person that you just poisoned to kill you, and there usually isn't a place to run off to while you wait for your poison to do its job if you're using poison in the heat of battle. The best use of it would be in an assassination attempt, where the assassin intends to stab the victim and run off, but in that case, why bother with close combat? All of the same sorts of methods could be used with a longbow or a dart gun, in which case the recipient of the poison and their friends would never even need to see your face.