2 weeks is, I believe, too small a period of time to go all the way from Patra to Romania/Bulgaria and reach the Italian Alps (that is if you want to do something else besides riding).

A feasible, and interesting, scenario would be to do Greece, a bit of FYROM, and then ride the Dalmatian coast north towards Italy.

If you choose to do that, you have plenty of interesting sites to choose from, and great routes to enjoy your trip. What do you prefer? Ancient ruins, Byzantine ruins, monasteries, lakes, mountains, citadels, national parks? What about roads? Highways, national roads, regional small roads (my favorite), mud tracks?

An interesting route would be:

Patra
Lefkada (you get to see an Ionian Island, it is connected by bridge to the mainland)
Ioannina
Meteora
Prespes Lakes
Cross into FYROM by NIKI border station
Visit Ochrid
Then head for Budva (Montenegro) through Tirane (Albania)
Dubrovnik (Croatia)
Split
Plitvice Lakes
Ljubljana
Italian Alps

This is just a suggestion, you have infinite choices, if you can be more specific I could give you a lot more ideas...

As for the when, late September and early October are the best for a bike trip in the region, before that it's too hot, and after you'll get lots of showers.. June is not bad either...

I doubt you can rent a car from an EU country, and then cross into non-EU countries (especially Albania, the only country that us Greeks get to pay extra insurance to visit with our own vehicles). If you stick to EU countries you could circulate freely, but not through the Balkans (not yet at least!).