As one Rome 2 player to another: it's a fair question. The game really incentivises you to "use up" your mercenaries before they have a chance to drain your coffers. Doubly so if you're playing as the Getae.

But this ties in to one of the key differences between the game and the actual ancient world. In the game, any unit you recruit or hire stays in continual active service until you disband it - possibly for centuries, if you're playing the grand campaign. Under these conditions, mercenaries are just not as cost-effective as native units, since they cost more every turn than their native equivalents. The only reason you'd ever consider paying the price is if your faction can't produce the unit's unique skills from its own unit tree (like if you're Rome but you want some archers).

By contrast, real ancient armies were raised to fight particular campaigns and disbanded immediately after. With the exception of some standing units (and the entire Roman army from Augustus onwards), ancient militaries did not tend to be on any kind of permanent footing like modern professional forces. Greek and early Roman armies were citizen militias, composed mainly of farmers who could not afford to be away from their farms for more than a few weeks in summer. Imperial armies like that of Persia were built up from local levies around core units of wealthy landowners. Even the armies of the Hellenistic Successor states were only professional in that their core units were typically composed of military colonists whose twin jobs were farming and soldiering. In peacetime, most ancient states possessed no more than a skeleton force - a few garrisons in key strategic places and perhaps some guard units in which the sons of the elite learned the ways of warriors.

There are several reasons for the ad-hoc nature of ancient armies. One was cost; keeping a large army in the field required a large amount of upkeep and a constant supply of surplus food. Most ancient states simply didn't have a financial system that could extract enough wealth to make this work. The solution was to keep the regular military expenses low, saving up to field as many men as possible when it really mattered. It's a general trend for ancient states to expand their standing forces as their finances improved, and for notoriously poor states (like Sparta) to possess no professional troops at all.

Another reason was that, in the subsistence economy that prevailed in most of the ancient Mediterranean, the manpower to maintain large standing armies simply wasn't available. Imperial Rome may have managed a vast professional army of 600,000 soldiers, but no other state had yet been capable of freeing that many able-bodied men from the constant obligations of farming and fishing to survive. Large numbers of men could be raised easily enough, but if they did not return home before the harvest - if their families had to do without the labour they could provide - there might be a famine. It was rare for rulers to keep their troops in the field longer than a few months, even rarer to keep them active for more than a single fighting season. When the fighting was done, the men went home. After that, the only ones who could still defend the realm, or besiege enemy cities, or campaign abroad, were those who were not bound by the need to grow their own food. The leisure class may have been happy to serve through the year, but they still had their estates and family interests to worry about.

You can probably see how mercenaries fit into this scheme. First, if you're only going to be able to pay for a small standing force, that force had better be the best that money can buy. Sources as early as Aristotle attest to the difference in skill between mercenaries and militia; as he put it, the fight between the two is like a match in the ring between an athlete and an amateur. Fully professional and loyal only to their paymaster's purse, mercenaries could be counted on to serve well as garrisons and specialist forces. Second, if you ever find yourself wanting to campaign out of season, or to protect your country when the people are busy with the harvest, mercenaries are the guys you need. Readily available at any time of year, they have no obligations at home and no rights as citizens that you need to respect. You can hire them and use them at need.

It's for these reasons that mercenaries were common in the ancient world. They were often used as garrison troops (like the Greeks in Egypt in the 6th century BC, or Iphikrates' peltasts at Corinth in the 390s BC), to campaign out of season or overseas (like Brasidas' mercenaries, who went with him to Thrace in 424 BC and still appear on the Spartan payroll six years later), to act as bodyguard units (like the mercenary life guards of Greek tyrants and Persian satraps throughout the Classical period), and generally to perform any task that the mass levy was ill-suited for. Mercenaries were also often used to supplement large armies, especially with specialist troops that were not supplied by the militia, but that was in part because it was in major campaigns that a state or ruler would invest the lion's share of its military budget.

On top of all this, rulers who weren't sure of the support of their own populations might prefer to build their armies out of mercenaries than to levy the people at all. Greek tyrants, frequently hated by the people they ruled, were especially known for this, and toward the later Classical period the armies of tyrants like Dionysios of Syracuse and Iason of Pherai are clearly pointing the way toward Philip of Macedon's professional standing army. These rulers supplemented their local forces with such vast numbers of permanently established mercenaries that they could do without their militia if necessary. The money for this came from plunder and taxes, often offering citizens the opportunity to pay for a mercenary's upkeep rather than serve in person. But even in their case, the number of men under their command was inflated drastically when they were planning a war, and when it was done, they came up with various schemes to get rid of them again.

In short, mercenaries were a valuable supplement to the typical ancient militia army, which didn't usually have the permanence (or training, for that matter) to take care of all of a ruler or state's military needs.

So how were they used in battle? I can't really speak for the Hellenistic era or Rome here, but for the Classical Greeks, it may not really be the right question. Pitched battles tended to be fought by large militia armies in which mercenaries rarely formed more than a minority; fighting those battles was something the militia could handle just fine (for the most part) and not where it needed mercenaries the most. In these cases mercenaries tended to be seen simply as a supplement to the main line. The remnamts of the Ten Thousand were part of the main battle line at Koroneia in 394 BC, just as they had been in the service of Kyros the Younger at Kounaxa 7 years earlier. At Mantineia in 418 BC, the mercenaries still known as "the men of Brasidas" (though he had died 4 years earlier) were part of the Spartan left wing, but not its crucial flank guard; that honour was reserved for the Skiritai. At the Long Walls of Corinth in 392 BC, the Argives made the mistake of deploying Iphikrates' peltasts on the right flank as though they were hoplites, and the enemy heavy infantry immediately put them to flight. At Kynoskephalai in 364 BC, the mercenary bodyguard of Alexander of Pherai fought around him as part of the Thessalian phalanx. In each of these battles, the mercenaries were considered neither better nor worse than their citizen counterparts, neither more nor less expendable, but merely a useful way to put more spears to the field.

I know of only one case where mercenaries were deliberately used as a forlorn hope. This was done without prior warning by the scheming Dionysios of Syracuse, when his army attacked the Carthaginians besieging Syracuse in 396 BC:

The cavalry and a thousand infantry from the mercenaries were dispatched in advance against that part of the Carthaginian encampment which extended toward the interior. These mercenaries were the most hostile, beyond all others, to Dionysios, and had engaged time and again in factional quarrels and uproars. Consequently Dionysios had issued orders to the cavalry that as soon as they came to blows with the enemy they should flee and leave the mercenaries in the lurch; when this order had been carried out and the mercenaries had been slain to a man, Dionysios set about laying siege to both the camp and the forts.

-- Diodoros 14.72.2-3

As you can see, the principle was not that the mercenaries were both expensive and expendable, and therefore could be sent to die without a second thought. Instead, the problem was that they were unruly and untrustworthy, and Dionysios thought he would be better off without that particular group. He therefore sacrificed a thousand men but kept the rest well in hand. Their sudden attack threw the enemy into confusion and therefore served a tactical as well as a personal purpose. Dionysios used the rest of his mercenaries to rout and destroy the Carthaginian army and fleet. In the aftermath, when he proved unable to pay the arrears for 10,000 of his hired troops, he offered them land in Leontinoi instead.

Now, there are exceptions to everything I've said here, and I'm sure a flair with more knowledge on Rome will be able to add a lot to my post. But I think this answers your question in a way that holds true in general, and certainly in the context of Classical Greece.