The way skill upgrading is explained here isn't the most straightforward way to teach someone how to calculate on their own.

The more general way to explain it is:
It's a chance that's calculated as a numerator X divided by a denominator Y (that is, chance = X/Y).

X is the sum of values contributed by all the weapons used as fodder here.
A R weapon contributes 10 times its skill level
A R grail contributes 20 times its skill level
A SR weapon contributes 35 times its skill level
A SR grail contributes 50 times its skill level
A SSR weapon contributes 350 times its skill level

Y is rarity multiplied by skill level.
For a R weapon, it's 5 times its skill level.
For a SR weapon, it's 10 times its skill level.
For an SSR weapon, it's 20 times its skill level.

- - - Updated - - -

UPGRADING SR SKILL LV

1 to 2 - 1x R
2 to 3 - 2x R
3 to 4 - 3x R
4 to 5 - 4x R OR 1x SRs2
5 to 6 - 5x R OR 1x SRs2
6 to 7 - 1x SRs2
7 to 8 - 1x SRs2
8 to 9 - 1x SRs2 + 1x R
9 to 10 - 1x SRs3
10 to 11 - 1x SRs3
11 to 12 - 1x SRs4
12 to 13 - 1x SRs4
13 to 14 - 1x SRs4
14 to 15 - 1x SRs4
15 to 16 - 1x SRs4 + 1x R OR 1x SRGrails3
16 to 17 - 1x SRs4 + 2x R OR 1x SRGrails3 + 1x R
17 to 18 - 1x SRs4 + 3x R OR 1x SRGrails3 + 2x R
18 to 19 - 1x SRGrails4
19 to 20 - 1x SRGrails4

SRs2 = SR + 1x R
SRs3 = SR + 3x R
SRs4 = SR + 6x R