How necessary is. . .

[doseyclwn]

How necessary is the core class pack to running a game?

[Zacchaeus]

Technically it isn't necessary at all. If you have the paper copy of the PHB (The Core Class Pack in essence) then you can enter all the character information, spells, feats etc manually. If, however, you want to save yourself hours of work and take advantage of the drag and drop functionality of character creation then it is a vital component. Out of all of the core rule books I think the PHB comes top of the list.

I should also point out that the SRD which comes free with FG does allow for drag/drop functionality and it does allow you to create characters. However much of the really nice detail has been left out of that. Things like Archetypes, feats, backgrounds and other stuff isn't there. So, again, I'd say the PHB is fairly essential.

It is of course, as always, your choice.

[Downhelm]

You can also purchase the starter adventure "The Lost Mines of Phandelver" and play with a group to see if you think it's something that you could continue to invest in.
The starter kit comes with some pre-generated characters and includes all the monsters and encounters you'll need for the adventure, it's a very nice starting point for DM's and Players alike who are trying to figure out it's worth, btw very worth it!

[damned]

Also the Basic Rules which are also included free on all licenses have less classes and races than the SRD does, but have more details and options for these than the SRD does...
I think Core Classes is the single best addon for 5e but you can get away without it if you accept the extra manual input or the lesser options from the Basic and SRD modules.