At the end of July I will be flying out of Canada to go to Greece. I do live in the United States and want to avoid any trouble flying out. Should I have difficulty at the border.
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To do this, you need to be admissible in Canada. One reason Americans can be denied entry is having a conviction for driving while intoxicated. No doubt, other crimes can also be grounds for refusal.
Add long as you are admissible in both Canada and Greece, and the border officers and the airline have no reason to suspect otherwise, you will encounter no difficulties.
Are you saying, you holding a US Passport, will drive from the USA into Canada, and from Canada catch a flight directly to Europe?
This will be no problem at all, in fact it is very common due to the air pricing differential on the US/Canada border.
Only possible problems I can foresee could be some kind of criminal background issue as @phoog states, or something that you are bringing in that's quite illegal in Canada (drugs etc.). It would have to be something fairly bad to cause you to miss your flight, as opposed to just having your meat sandwich confiscated (anyway, it's more the U.S. that does that). Probably Greece has similar restrictions to Canada on most things for what you are physically carrying with you so unless you're packing your submachine gun in the car trunk and planning on leaving it at the airport parking you should be fine.
Generally this kind of thing goes very smoothly (though leave plenty of time for formalities at the border- delays can be tedious and you don't want to start your journey all frazzled)- I have done this many times going the other way.