What you're getting into here is largely a historiographical issue: how do modern historians explain how the Romans acquired such a huge and unprecedented (in Western Europe, at least) empire?

As you might expect, the views on this question have varied a lot depending on the historians' region of origin, and perhaps more importantly, the era in which they were writing.

(N.B.: I'm simplifying the debate here for clarity's sake. Dissenting voices existed in all eras of historiography.)

In the 19th century and before, most historians writing about the Roman empire lived in European nations with empires of their own, and as such tended to be favourably disposed towards the Romans. They'd write in terms of "civilising missions" where the Roman conquest was, supposedly, a great benefit to the people they subdued, and "defensive imperialism," where the Romans were more or less forced to acquire their empire because people kept attacking them. (Much like the British at times claimed they acquired their empire "in a fit of absentmindedness.")

Post world-war II, this kind of view became very unpopular. As decolonisation took root, history started to focus more on the perspective of the conquered rather than the conquerors, and a focus was put on the many hundreds of thousands who were killed by the Roman armies, by the injustices and brutalities that would occur in provinces under Roman rule, and the loss of native cultures and ways of life. This type of perspective on the Roman empire has continued to develop until the present day, although outside of academia the old view of "Romanisation" and civilising missions continues to persist. (This article by Jane Webster offers important insights on this.)

Anyway, it is in this era that among some historians the explanation "The Romans acquired their empire because they simply were the most aggressive and warlike people around" took root. The work of William Harris (War and Imperialism in Republican Rome 327-70 BC) is of particular note here, where he argues that glory and profit were the main motivations for Rome's conquests.

This view too has shifted. Whilst it's still not done to be an overt cheerleader for imperialism, Roman or otherwise, (and with good reason, I might add) the most recent generations of historians have started to examine Roman imperialism more in the context of their own times.

The conclusion I most agree with, summarised very crudely, would be that yes, the Romans were very aggressive, imperialistic, and at times brutally ruthless. But not notably more so than everybody else. The success of the Roman empire cannot be explained by their aggressive and warlike culture, since everybody had such a culture. In antiquity, war was the rule rather than the exception, and although war did need justification, it did not need an excuse and was mostly considered to be a good thing. The Romans success is then explained by exploring how they were better than their neighbours both at mobilising their people for war and at integrating defeated peoples into their empire.

So in this one sense you could say they were the most warlike people of their era: the Romans, at times, mobilised an unprecedented percentage of their people for war. We see this most significantly during the Punic Wars, when at one point an incredible 25 Roman legions are in the field. Depending on how you calculate the population of Rome and Italy at the time (a thorny issue - PDF warning) this means that the scale of mobilisation in the Roman world was unmatched until either the time of Frederick the Great in 18th century Prussia, or until the time of the French Revolution and the mass-conscription that followed.

As for their empire: I'd say that some of it was acquired "defensively," i.e. in response to outside aggression. Some of it they took because they could, and much of it was conquered by ambitious individual generals and governors who weren't held in check by the weak systems of late Republican government. On balance, the great majority of territory was taken in aggressive wars, albeit mostly opportunistic ones driven by individual actors rather than state policy. However, from Augustus' time onwards the Romans became much less aggressive and far more focussed on defending what they had than on acquiring new territory.

That said, all these comparisons make the most sense when comparing Rome to their Italian neighbours, the Greek poleis, or the Hellenistic kingdoms, or the Carthaginian empire. When comparing the Romans to, say, less centralised and urbanised British or Germanic peoples, you may find that the nature of violence in their societies changes, and the question of "who is the most warlike?" starts to depend on what aspects you look at.

For example, the Romans would be far more likely to gather up an army to invade and annex distant overseas territory (in some periods of their history) than any given British tribe would. The Romans would be far more likely to sell entire peoples off into slavery. That simply isn't the kind of warfare that took (or seems to have taken, our sources are thin) place in those more peripheral territories.

On the other hand, you would find more endemic low-intensity warfare, where raiding and skirmishing between neighbours is more the exception than the rule. Likewise, internally you might find more blood feuds and related inter-personal violence in a tribal society, but you might find more oppressive taxes, brutal corporal punishment and corrupt or arbitrary governance in a poorly-governed Roman province. More recent in-depth demographic research based on archaeology - PDF warning, meanwhile, suggests that Roman rule did not exactly improve the quality of life for most of its people, and may have had a negative impact in some places.

In other words, trying to compare Germans to Romans on questions of "warlikeness" is something of a case of comparing apples to oranges.

Finally, since the reading you were at compared the Romans to the Han empire, I'll just say that I know far too little about Chinese history to comment, but Walter Scheidel has written a slightly older but still interesting article - PDF again comparing the rise of Rome to that of the Qin and the Han, which may interest you if you're looking for some solid academic insights of digestible length. (I suspect the reading you went to may not have been quite as solid.)