There's a lot of toxic metals. Most are toxic in fact (including iron technically, if you get enough of it). Almost all of them have issues though:
- Mercury: One of the first metals many people think of when they are thinking of poisons. The problem here is quite simply that even those mercury alloys that are solid at room temperature (known as amalgams) are generally not hugely toxic (they were even used classically for dental fillings). In fact, most of the mercury compounds that are toxic are entirely non-metallic (see dimethylmercury for a particularly nasty example), and therefore not very good for swords. The fumes from mercury are, however, rather toxic (they're a slow acting neurotoxin), so maybe you could coat your weapon in mercury (though that would put you at risk too).
- Lead: Is a very slow acting toxin, and does not lend itself well to usage for a sword. It's also not very readily absorbed by the body in pure form, with most of the toxic forms being salts (white lead for example) or organic compounds (like tetraethyllead). It's also not good for a sword, it's way too soft, and it tends to soften other alloys (steel with a high lead content has a significantly lower shear strength than steel without much lead in it).
- Cadmium: Is also mostly a slow acting toxin. Most of the same issues that lead has for this usage are also problems with cadmium. The biggest difference is that it would be a whole lot more dangerous to the swordsmith to make a sword out of cadmium, because the fumes and dust are far more dangerous than those of lead. Interestingly, cadmium is rather corrosion resistant, so if you were to go with this, the sword would actually not need to be cleaned as thoroughly to keep it in good condition.
- Thallium: Most people probably haven't heard of thallium outside of chemistry class as it's not used for much. It's located right between lead and mercury on the periodic table and is actually significantly more toxic than either. It's interestingly one of the few metals which are actually sufficiently toxic in pure form to be dangerous to handle and might work here. Unfortunately, it's not very practical for this because the swordsmith would most likely receive a lethal dose well before he finished forging the sword, and it's also very soft and has a low melting point (just like lead), which means that alloys incorporating behave similarly to those which incorporate lead. Somewhat notably, thallium and its salts were a favorite means of political assassination in ancient Rome.
- Osmium: Like thallium, most people won't have heard of this one much outside of chemistry class. Osmium might be the closest to what you're looking for. It naturally oxidizes in air to form osmium (VIII) oxide, which is insanely toxic (concentrations so small you can't tell it's there can cause pulmonary edema and permanent blindness) and also sublimes readily (converts from a solid to a gas). Osmium is also the densest naturally occurring chemical element known to man, so anything made of it is heavy (good for a weapon if used right), and it's also rather durable too (it's used in small amounts in alloys for fountain pen nibs for this reason). The big issues here are that just like thallium, it's liable to kill the swordsmith well before he finishes the weapon, and, more importantly, the weapon would just as dangerous to the one wielding it as the intended target (because of the fumes).
- Chromium and vanadium: Both of these are probably already in your steel in small amounts. Both of them have a number of compounds that are very toxic (vanadium more than chromium). Unfortunately, none of these compounds can be produced in a weapon (they're all salts in some way) in any way that wouldn't make the weapon significantly less useful (they would compromise the structural integrity of the weapon).
- Nickel, copper, cobalt, silver, scandium, and yttrium: Similar story to chromium and vanadium, just not likely to be in your steel already.
- Zinc: Zinc is actually an essential nutrient. Like most metals which are essential nutrients, getting too much is very bad for you. Overabundance of zinc in the body causes all kinds of nasty things, interfering with copper and iron uptake, destabilizing cholesterol levels, and acting as a mild neurotoxin with permanent effects. The problem with using it for something like this is that it's the ionic form, not the metallic form, that causes problems and is readily absorbed. In fact, most of its alloys that are actually practical for swords are pretty much entirely non-toxic. A pure zinc stiletto might be viable for this if you stab someone in the stomach (such that it ruptures the stomach), because stomach acid dissolves zinc, and the resultant zinc chloride solution is really corrosive, but such a weapon would effectively be a single-use affair.
- Antimony and arsenic: Not technically metals, though they are found in measurable concentrations in many old steel alloys. Both of these are reasonably fast-acting toxins in their oxide forms and oxidize readily. However, proportionate to how much you can reasonably put in a sword the size you're describing without making the steel useless for a sword, it takes a lot to actually kill someone. They are however persistent toxins (that is, if you get a bit of arsenic or antimony oxide in your body, it's going to stick around for a long time), and do have effects prior to reaching lethal levels (they make the victim tired, because they interfere with energy storage in their cells).
- Selenium and tellurium: Similar story to antimony and arsenic, except these generally aren't found in even the oldest steels, and the mechanism of toxicity is somewhat different.
- Aluminium: Believe it or not, aluminum is actually toxic to most living things. It's unfortunately impractical for what you want for a lot of the same reasons that aluminum foil doesn't poison people, it forms inert compounds very readily (aluminum oxide is non-toxic and extremely unreactive, and forms readily in air), the metal itself isn't well absorbed by the body, and rather large amounts are needed to cause significant effects.
- Beryllium: First off, beryllium is impractical for a sword, it's too light and brittle. There are some other issues with using it for this though, most notably that it's not really all that toxic in its pure form (you should be noticing a theme here by now) or when alloyed (beryllium copper is a good example of an alloy that might work for a sword, but it's generally considered perfectly safe), with the compounds being the biggest issue (except see below). It's also not easy to alloy with most metals, difficult to work, and, most interestingly, is primarily an issue in the form of dust (which has similar long-term effects on your lungs to asbestos).
- Lithium: Lithium has similar issues to beryllium, but deserves separate mention form the other alkali metals because it also has some of the same issues that aluminum does (namely, it readily forms inert compounds).
- Rubidium and cesium: Both of these are really nasty slow acting toxins if ingested in the form of water soluble salts (for example, chloride or nitrate salts) because the ions interfere with biological processes that normally use sodium or potassium ions. Both of them are really nasty explosives if exposed to air (they react with the moisture in the air just like potassium and sodium react with liquid water, just way more so). THey're also damn near impossible to isolate in their pure form by any means other than modern technology. Given these constraints, they're impractical for what you want.
- Strontium and barium: Same issues as rubidium and cesium, except they're not really explosive in air.
- Lanthanide metals: All of these are toxic to some slight degree. They also have physical properties that make them unsuitable for use as a blade and are insanely difficult to isolate (to the point that many of them were not isolated in their pure form until the later half of the 20th century).
- Radium, uranium, and other radioactive metals: All radioactive, so they violate the requirement that it does not harm the user. Also, actual toxicity other than the radioactivity is not well characterized.
- Polonium: Polonium is insanely radioactive. So much so that less than one microgram (one millionth of a gram, that's about 1/60th of the size of a grain of table salt) is enough to kill someone due to acute radiation poisoning. Handling it is dangerous to the point of being almost suicidal. It's also really rare and extremely unstable, so the likelihood of getting enough to do anything practical with in medieval times is essentially nil. Given all this, while it would be very effective, it's also highly impractical. I've mentioned it here separately from the other radioactive metals simply because it's so much more dangerous.
I've refrained from mentioning many of the transition metals above, as they're either not reactive enough to be meaningfully toxic (titanium, platinum, palladium and iridium), too hard to isolate or reasonably work with in medieval times (tantalum, hafnium, niobium, manganese, tungsten and all the ones whose name begins with 'R'), or are just plain impractical for a sword.
So, overall, you're not likely to find a metal that can do what you want just based on metal alloys. There is also the issue of exposure. Cutting someone does not allow for much exposure, because the blade is kind of supposed to remain in one piece without losing much (if any) mass. Even regular blades coated in poison are not particularly reliable in real life, as reliable delivery of the poison is still difficult even if it's a liquid.