I was thinking of creating an App for studying Kanji and for this I would like to use the Kanji stories of this website licensed under https://creativecommons.org/licenses/by-nc-sa/3.0/
I was thinking of a free app for like the first 300 kanji and a paid (cheap) app for the rest, however I don't intend to make money from content I did not create (the Kanji Stories). As such the stories will be available for everyone under the same license.

My question is if I'm allowed to use these stories to improve my app as I won't use this part to make money (and make this freely available) and I only charge for the part which I coded myself and other features the app has?

p.s. Of course the right users and this website will be credited.

Have you read the attribution requirements?
http://kanji.koohii.com/about/license

Also I'm not sure I follow your reasoning. Since only the first 300 kanji are free, that means any stories you use beyond this point is commercial use, don't you think?

Of course the attribution requirements will be followed.

That's indeed true, that's why I will do something like include all stories in the free version (.txt files don't take much space) or provide a download link on a website.
To clarify, in this case the files are put on the storage of the phone as .txt files and therefore approachable by PC or mobile file explorer. I'm still at the beginning of creating the app and therefore I'm not certain how I will implement this yet.

[ファブリス]

Kanji Stories by Phi Coding, for iOS.

He got in touch with me. The app features two stories per character that the author selected, and other stories are available with the "More Stories at Reviewing the Kanji" link, which is a very responsible and fair way of doing this.

As far as I know, the stories provided can be edited and /or replaced with your own.

The app is a complete package, the non commercial license I chose is to specifically discourage people trying to make money of the user contributed content of this site. Unless the author got in touch and has my express permission, as is the case here. What matters to me is the author's motivation, and how effort they put into their apps.

It's not ok to include all the stories from the Reviewing the Kanji in your app, no matter how many mentions you make of the site. This is simply taking a lof of value away from a site I put a lot of personal effort and motivation in building and maintaining over the years. The audience is all I have to keep me going, along with occasional donations and affiliate links. Take that away and you take away my motivation to keep the site and forum going for years to come.

Ideally an app featuring stories from this site, just like Kanji Stories, would feature a personal selection of stories (via your own "exported stories"). And then feature a link or button that lets the user view "more stories". Since this requires the user signing in RevTK, I think that is a fair way of adding value to your app, while potentially bringing back some of your users to sign up on RevTK.

If the app features Remembering the Kanji keywords, you should also seek express permission from James Heisig.

Another way to approach this, is to implement a way to import the user's stories as exported by the RevTK site; In that case the app is more of a companion to the site, and as such I would be happy to promote it.

Thank you for your reply Faburisu.

I'm not sure of the details of the implementation, but the current idea is to only include 1 story per Kanji. I would be happy to collaborate with you so both our wishes are being honored. I will also look into the possibility of linking the App to the user's personal stories on this website. When I know more specific details about my App I will get in touch with you with the details.

Thank you for your hard work you put in this website.