If you happen to obtain kakuzu 4* in the near future, when you go for his skill trial, make sure you get the gold +1 for his double standard passive first. Only then should you start using protection vials to keep that skill and hopefully unlock the rest of the skills like his scale, standard attack, mystery etc.

It's quite simple. Breakthrough involves obtaining stronger versions of your current skill. For that, you would either go with the RNG method of using vials OR you go with the long way which is to save coupons to gather additional frags to exchange for books (once you've upgraded your ninja to 4* ofc). Each method has it's benefits and risks.

Experimental vials and protection vials:

With luck in RNG, you can spam your vials and see if you obtain stronger versions of your current skills. Although the RNG is quite poor (treat the RNG like purification), you have the slim chance of unlocking strong skills without having to wait months to gather coupons for skillbooks. However, you'd require many vials and protections and ofc there's no guarantee you'll unlock the desirable skills. You might even run low on vials and protections.

Skillbooks:

Although this is the longer term method, by saving coupons and exchanging them for books you're guaranteed to unlock the skills you want. However, it takes a * lot of time and is quite a waste considering most players are unlocking skills through vials. The only benefit over vials is that you're guaranteed to get the skills you want.

Personally i'd just rack up on vials and blast my way through with RNG. This is just something you'd want to consider for the future. Good luck farming Kakuzu.