How many "kills" would a legionary roughly have after fighting in the army?

That number is ridiculous.

If that number were true, each Roman legion would kill 1.5 MILLION people for each generation of serving soldiers. Considering Rome in the age of Augustus had 25-28 legions and the entire empire has been estimated to have 56.8 million inhabitants, this would rapidly have exterminated the entire world population.

(Edit: Before it was edited, the OP posited 250 kills -per legionary.- That's what this reply is based on.)

But it's hard to come up with a more realistic answer. This question is really difficult to answer and would vary wildly depending on the individual, the location and the timeframe.

A soldier serving in the 10th legion with Caesar in Gaul and then the Civil Wars might live through as many as 10 or more major battles. But a soldier serving in the early second century AD along the lower Rhine frontier might spend 20 years under the standards and see no big battles at all.

Even if we somehow know which battles a legionary took part in, we have no idea how many – if any – enemies the average man would have killed. Goldsworthy (and other recent military histories seem to hold to this same number) holds that victorious armies suffered up to 5% casualties, but usually far less. This is then typically held to be the rate of attrition during a battle. The defeated side suffered far more casualties, (30-50%) but those would have been disproportionately inflicted by pursuing cavalry. Assuming even numbers and assuming an even distribution of kills, an average soldier would then only have a 1 in 20 chance of killing anyone at all during a battle, but would have a much bigger chance of killing if his side won. (But that would be a false assumption. Elite and naturally aggressive warriors, who'd be in the front ranks, would be doing most of the killing. In Rome's armies the centurios would probably take the lead. So you can't just calculate an average and expect it to reflect reality.)

The problem with this approach is that casualty figures for ancient battles are notoriously unreliable. The victor habitually exaggerated the size of the enemy army to increase his own glory. Historians have been arguing passionately since the 19th century (at least) about how realistic the figures we have are. Some roughly accept what the ancient authors tell us, some seek to propose drastically different numbers. Suffice it to say this throws all calculations out of whack.

Moreover, even if we somehow were to determine with some semblance of accuracy how many enemies the average Roman killed in a big battle, we're ignoring some very important aspects of military life. Armies fought skirmishes, sieges, raids. They sacked towns and villages. They might kill enemies (Or civilians) in any or all of these.

There's another way to try and approach this: if we look at Caesar's Gallic wars, and know the total number of casualties and the total number of soldiers serving, we can simply divide the one by the other. If I were to take Wikipedia's figures of 1,000,000 Gaul casualties (mostly civilian) and 120,000 soldiers serving in Ceasar's armies, we get an average of 8.3 kills per soldier. Though again, as I pointed out above, it would actually vary wildly between individuals.

If this is true, it would be a statistical outlier, because of the extraordinary bloodiness of the Gallic wars. A soldier serving a century earlier or later, or in a different theatre of war, would not get anywhere near that figure.

Edit: and of course this number completely ignores the fact that soldiers rotated in and out of the army all the time. Some died or got injured, others joined up, new legions were raised, etc, etc. To get a true average we'd need to know the total number of men that served in Caesar's army at any point in nearly a decade of war. Presumably, this means that even if Wikipedia's figures are somewhat correct, the average kills-per-soldier would be much lower. /end edit.

In conclusion, I am confident of the lowest possibility in the range: Zero. It is quite possible a soldier would serve an entire career in the Roman army without ever killing anyone. This is even possible if he takes part in battles, though it's likewise possible a soldier might serve for 20 years without drawing a sword in anger.

The high mark might be quite high indeed. We have anecdotal evidence (from Caesar, say) of Romans killing several foes in a single battle. A cavalryman who participates in several battles where he spends hours pursuing routing enemies and sacks several towns or cities, and also fights in skirmishes (cavalry would more often than infantry) might indeed rack up a very high body-count. But I have no idea what a realistic average figure might be.