Before you look at capacitors, look at the pots. If you're using a '62 reissue P-bass pickup, it was designed to work with 250 Kohm audio taper CTS pots.
Then from there...some folks like more brightness, and use 500kohm pots. Others like a modified taper, and will bridge the 500kohm pot with a 1meg resistor to bring down the overall value to @330Kohm with a modified taper.
The most common mod to the control circuit is to use a 250kohm linear taper pot as the tone control pot for a finer degree of adjustment.
Then after all that you finally get to capacitors: the current standard for a bass is a .047, and some folks have golden ears that claim to tell the difference between ceramic and other types, I prefer to save my $$ and get one from Radio Shack, although on occasion, if they are on sale, I'll use a Sprague Orange drop. Others here on the forum will have their personal favorites. I also like a .033 instead of a .047 to get more mids; Leo started out with a .1 in the circuit to get more bass. Others use a .068 for more of a dub tone.
Your mileage will vary.
Bottom line: if you want true vintage tone, use 250 Kohm CTS pots and a .047 tone capacitor.
Oh - and installation - by all means, when you bend the leads a half turn around the pot lugs and solder, then clip the remaining lead ends so they don't short something out.