[MNpoker]

A discrete function will never become perfectly normal if the underlying discrete function is not normal. It will however become MORE normal as you increase the number of independent trials.

For example say your indivdual hand distrbution is:
- 1 90%
+ 11 10% of the time

If you run 100 trials there is still a .1^100 chance you will have won 11 * 100. (you win every trial) There is a zero chance you will have lost 11 * 100 (in fact your largest possible loss is 100).

So what you need to decide first is how close to normal do you need the aggregate results to be to be considered normal?
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On the changing parameters. (Good table, playing well, etc.) This will be VERY hard to empirically estimate, and will make the calculations more tedious.
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Some methods that may be usefull in seeing how many hands you need to approximate a normal function would be:

Get every hand you have and put it into a discrete curve with the values as percentages. There won't be that many buckets. (You can lose up to 12 BB, but probably never have, and win up to 108BB <-- That I'd like to see). Most likely your range will be something like -8BB to +30BB with an interval size of .25 BB <-- The small blind.

Now you can either
A) Run X simulations off that curve and see if the result is 'normal'

Or

B) Do a FFT on the curve with X convolutions(?) and see if the outputs are 'normal'.

If the results are still to skewed to be called 'normal' increase X.
The FFT method has the advantage it gives the 'true' distribution but it's a pain to do (I would guess you will need somewhere in the neighborhood of 5,000 convolutions(?) and you certainly won't pull this off using FFT in excel)
The simulation method if you get a 'normal' result you should probably still run it about 10 times (minimum) to make sure you didn't just get a 'normal' simulation.