It's debatable whether Lydian/Persian coinage can really be called "Greek", although the style is very clearly influenced by Greek coinage. The first coinage of Greek proper was of Aegina, around 620 B.C.E, although this is simply an estimate based on archaeological findings. It's less clear when Greek coinage ends: Diocletian's currency reforms are a good point for the issues of Greece itself, but the hellenic issues of Bactria (west India) and so forth also have to be taken into account. Other coinages gradually became less Greek in style: for classification purposes, anything after 450 C.E would probably not be considered ancient.
The Bactrians had gone by the time Diocletian was around. You are right about the begining of greek coinage being a gray area though. However, I think if one wanted to start a comprehensive greek coin collection, you couldn't really go wrong by buying a tetradrachm of Diocletian (or Dioclese) and a Electrum stater of Lydia and then endevouring to fill in the the gap between them....
I'll accept the span suggested but the whole idea of collecting Greek coins is really faulted. Ancient Greece wasn't a place with borders but a culture that had a common language and an attitude that those who didn't understand that language were in some way defective. When 'Greeks' travelled to Italy, Spain, Africa or India they expanded the 'Greek' world. Whether the people there thought of themselves as 'Greek' is a different matter. Coinage expanded the Greek influence even farther than culture. Places that were very un-Greek issued coins with Greek legends or styles. Trade routes in Africa (Aksum) and Central Asia made Greek coins a fixture in places that never heard of Athens. Coin collectors force things into neat little categories. In my way of looking at things, coins started in three places. The Chinese cast metal in shapes including the round ones with square holes that became the standard. Indians cut flat metal and hit them with punches (often 5) with various meanings. That system lasted a few hundred years making it least successful of the three. Greeks took more or less round lumps of metal and hammered them between dies cut with a design. In the end, the Greek answer to coins took over more and more of the world to the point that it is hard to find a modern coin that is not a little 'Greek'. Where we draw the lines is wholly a matter of opinion. I'll accept the span suggested.
It ends in different times in different places. In Gaul, once Greek denominations like the obol and stater were changed out for Roman ones like the dupondius and denarius. In Egypt, I'd say that once Augustus altered the monetary system and introduced the billon tetradrachm, Greek coinage had come to an end. In Asia Minor and the rest of the east, it ended when individual cities ceased minting their coinage (and not the big, Imperial-style mints like Antioch). But, there is one issue of Greek-style coinage that presses beyond Diocletian's reform. Bosporan coinage, a bit of a hybrid between Greek and Roman, was struck until the death of their last king, Rhescuporis V, in 343 AD.