In the lesson called vowel rules, at the end there is this sentence

kedi-in-ki-den -> kedi-n-in-ki-n-den -> kedininkinden (from the cat´s)

´in´ is a suffix that gives genitive meaning and ´den´ is a suffix means ´from´, what about ´ki´?

ki means for ex,
your --> yours
senin --> seninki

ex: benim kitabım burda. seninki nerde?
my book is here. where is yours?

But i already have the suffix "din" that gives this meaning so why to use "ki"?

It emphasises it.

i.e. your book (rather than my one)

Or is like the use of "...´s one" in reply to a question

Which shirt is nicest?
Ahmet´s one.

Kedi: Cat
Kedinin: the cat´s

If you use this you have to add another noun:
e.g. Kedinin tüyü: The cat´s fur

Kedininki
The one which is the cat´s

Which bowl is smallest: the dog´s or the cat´s?

Kedininkinden: This now has a -nden ending that goes with the adjacent verb (the -n- is used as a buffer because -ki- ends with a vowl)

e.g.
Köpeğiniz hangi tastan su içti?
Which bowl did your dog drink from?
Kedininkinden içti.
He drank from the cat´s one.

Thank you very much, it is very clear now