It's the same as spending Karma on anything else in the game.
If you started with Pistols 1, and you wanted to get yourself to Pistols 6... you have to pay to raise it up to 2, then pay more to get to 3, so on and so on. You cannot simply go from "which end to the bullets come out?" immediately to shooting the wings off a fly at a thousand yards.
The wording of the rules on Core pg 105-106 are piss poor (SNAFU) but the chart on pg 107 makes it (kinda) clear.
Starting rating 1, raising up to 2 costs 10... or raising up to 3 costs 25... or raising up to 4 costs 45
Starting rating 2, raising up to 3 costs 15... 10 + 15 = 25
Starting rating 3, raising up to 4 costs 20... 10 + 15 + 20 = 45
So you're going to pay (5x2)+10... then (5x3)+10... then (5x4)+10... for the grand total of 150 Karma at Depth 6.