The short answer is: 4. The Daleks, the Last Great Time War, Parting of the Ways, and "Journey's End".
Most cases are ambiguous, so different answers are possible.
At the end of their very first appearance (The Daleks), they all die. We later learn that this city wasn't the only place where there were Daleks, but there's no indication in the episode at the time that such a thing is true.
In their last classic-series appearance ("Remembrance of the Daleks"), the Doctor tricks Davros into destroying Skaro. However, this time, there's no reason to believe that killed all of the Daleks (despite what some fans, and at least one writer, believed at the time), so it doesn't count.
the Doctor believed the Daleks were all destroyed in the Last Great Time War (which happened off-camera, after the TV movie, and all of the EDA novels, but before the new series). It later turned out that at least some survivors (the lone Dalek, the Emperor, the Cult of Skaro, and the Daleks inside the Genesis Ark). The viewer didn't find out about this until the Doctor encountered the lone Dalek and saw him die (Dalek), so you could call that the destruction instead of the War.
The Emperor created a whole new race of Daleks out of humans, but they were all destroyed by "Bad Wolf" Rose (in Parting of the Ways). That's 3.
The Cult of Skaro opened the Genesis Ark, freeing all of the Daleks inside, but they all got sucked into the Void (in "Doomsday"), and only the Cult escaped. Since we saw them escape, this doesn't count.
The Cult of Skaro began creating new Dalek-human hybrids, but they ended up destroying each other (in Evolution fo the Daleks), and only Dalek Caan escaped. Again, we saw him escape, so this doesn't count.
Caan broke through the Time Lock and restored Davros to the universe, and Davros created a whole new race of Daleks from his own cells. They were destroyed by DoctorDonna and the Meta-Crisis Doctor ("Journey's End"). One damaged ship with a handful of Daleks escaped, along with the Progenitor Device (which probably survived the War, but it's not entirely clear), but as far as we and the Doctor knew, they were all gone.
Those escaped Daleks tricked the Doctor into activating the Progenitor Device, creating a new race of Daleks who escaped into the future ("Victory of the Daleks"). They haven't been destroyed. (In fact, part of the reason for this episode was to get the Daleks back out into the universe so they didn't have to keep coming up with explanations for why they weren't actually all destroyed.)
The Daleks' first attempt to rebuild their empire on Skaro was erased from history by the Doctor and Amy (City of the Daleks), but we know that wasn't all of the Daleks; they're still out there.
Also, even without the Time War, we can assume that at some point in history the Daleks would have died out. If nothing else, the universe will eventually end. (We know this from real-life science, and "Logopolis", among other episodes, confirms that it's true in the Whoniverse.) In "Utopia", humanity (and "futurekind") are implied to be the only living things left anywhere, but this isn't certain. A comment by Rodrick in Parting of the Ways implies that their end might have been a few thousand years before 200100, although that's definitely debatable. A couple of novels have scenes in a far future where there are no Daleks (e.g., in Father Time, they seem to have been replaced by a humanoid race that the created called the Klade). Other non-TV sources imply that after the Third Dalek War (somewhere in the 5th millennium), at least humans thought the Daleks were destroyed. Whenever it happened, it presumably happened at some point. But I think it's a stretch to count this as another destruction.
"A comment by Rodrick in Parting of the Ways implies that their end might have been a few thousand years before 200100, although that's definitely debatable." In the same story, Jack Harkness said something along the lines of "One moment, they were the greatest menace in the universe, the next, they'd vanished," and the Doctor replied, "They went off to fight a bigger war," meaning the Time War. To Roderick and company, they'd disappeared and he, at least, thought that was forever -- but they hadn't been destroyed.
As I said above, most cases are ambiguous and debatable, so just proclaiming the number 7 doesn't do anyone much good.
I didn't count Evil of the Daleks, because at the end of that, they're not destroyed, just (apparently) no longer a threat to the galaxy. (They're locked in a civil war that the human-factor-infected Daleks seem likely to win. But you could argue that those wouldn't really be Daleks (and the Emperor would probably agree with you).
Also, counting the LGTW and Dalek separately seems a little odd. One lone Dalek dying doesn't really feel like an extinction—and if you look at the dialog at the end, the Doctor talks about it as the end of the Time War ("I win, how about that?"), not a separate destruction.
I Am a Dalek is trickier—it's a lone Dalek again, but nobody treats it as the end/continuation of the previous destruction, so you could count it separately. (And I forgot about that one.)
The question is how many times were the Daleks thought to be destroyed? the Doctor thought them all dead at the end of the Time War, so that's one. Then when he found that lone Dalek in Dalek and that one was destroyed, he thought them were then all gone so that's two, continuation or not, the Doctor thinks both times were the end of the Daleks.
"The Evil of the Daleks" I'd say counted. It wouldn't exactly be "The Final End" if the Daleks were only changed instead of being destroyed. Other sources like The Dalek Handbook state "This, the Doctor believed, would be the Daleks' final end" which, again, doesn't sound like only the end of their threat. Both the Handbook and a source given in Doctor Who: Battles in Time magazine's Dalek Wars section give the heading of this event "The End of the Daleks", and the klatter source claims the two Dalek sides destroyed each other and only rumours went around that the Emperor survived. I'd say they were thought to be destroyed there. I believe it's also widely considered that thios story was thought to be the Daleks' end, since it was originally planned for it to be the final Dalek story.
I Am a Dalek is a trickier case, which is why I put "likely" there. That story doesn't see the return of a surviving Dalek but the creation of a new one. But the Doctor wouldn't really expect to see the Daleks again after leaving, so I don't see why not.