Greek colonisation tends to be a bit exaggerated, partially due to the way in which it's portrayed in maps. Take this image, for example. While it's true that Greeks colonised those areas, it wasn't the sort of colonisation which later European empires carried out, where whole regions belonged to them. In reality, it would have looked a bit more like it does in this image (warning, 2800x1600), where the colonies were primarily localised city-states (much like Greece proper at the time). Robert Bonner has a passage on this in a journal article he wrote on the topic:

It is not easy to determine how much territory a Greek colony controlled beyond the wall that guarded the promontory upon which it was usually built. There was a considerable stretch of territory in the immediate neighborhood which was more or less effectively controlled by the city. Beyond this lay the friendly native communities and villages corresponding roughly to the modern sphere of influence. At suitable points in these territories, trade centers were established which were capable of warding off sudden raids but not sustained attacks. Naturally a seafaring people would select spots along the coast for the establishment of these so called Χωρία. These depots and the territory surrounding them constituted the territory (Χώρα) of the city. (...) It is not unlikely that the territory of a colonial city was comparable in extent to that of a city state in Greece.

Given this, while there would have already been inhabitants in a lot of the areas which the Greeks colonised, finding an uninhabited area to set up a city wouldn't have been too difficult. Indeed, some places (see Sicily on the second map above) had several different settlements from different colonists - the Phoenicians were the other main colonising group in the Mediterranean, and despite being a bit earlier than the Greeks, Phoenician colonies remained strong and eventually grew into the Carthaginians, who themselves colonised much of the western Mediterranean (and by some accounts, Atlantic coast of Africa) and rivaled the Romans in the Punic Wars. I'm digressing now, but the point is that the Greeks weren't the only ones doing the colonising, and in many instances they were colonising places which already had colonies (the Greek colony of Syracuse and the Phoenician colony of Ziz, (Panormos / Palermo) both on the island of Sicily, are both believed to have been founded in the eighth century BC, around the same time).

In terms of how colonies were actually founded, it varies quite a bit depending on which colony and which interpretation. Talking about the Greek colonies in the region around Panticapaeum, former director of the Panticapaeum Archaeological Expedition, V.D. Blavatskii proposed that the introduction of Greek culture and foundation of Greek colonies was a gradual process, brought about through trade (and piracy). As explained by Thomas Noonan in an article on the subject:

Blavatskii tried to demonstrate that all early Greek colonies of this area de- veloped from emporia in three stages: 1) sporadic visits by Greek merchants-pirates; 2) the development of trade ties between Greeks and native tribes which resulted in the foundation of temporary and then permanent emporia; and 3) the transformation of these emporia into cities of the polis type.

There are other arguments about foundation of colonies (such as overpopulation in home cities), but I can't find enough source material to include them at this point (and the post is getting way too long). I know I only answered part of the question, but I hope it's enough!

Sources:
Robert J. Bonner, Greek Colonies and the Hinterland, The Classical Journal, Vol. 20, No. 6 (Mar., 1925)
Thomas S. Noonan, Origins of the Greek Colony at Panticapaeum, American Journal of Archaeology, Vol. 77, No. 1 (Jan., 1973)

I wrote a paper as an undergrad on the motivations for Greek colonization in the Western Mediterranean. You seem more interested in the "how" than the "why" though, and more focused on the Eastern half of things. But I can post a link if that interests you. Mostly I discuss the older theory that Greek colonization in the Western Med. was driven by the motivation for metals, and some newer scholarship contradicting that.