I have this game. It's quite good. (although I use kakitorikun much more.)
However... it does seem aimed at Japanese adults who want to improve their handwriting. So it won't really help you learn readings or really introduce characters in a systematic way (that I can discern anyway). Although there is a sidegame where you go through all the kana and then the kanji in grade school order. It can also be hard to get into if you can't read kanji very well.
Primarily it helps you write kanji properly. ie well balanced. It won't accept incorrect stroke orders, or incorrect stroke counts. It gives a lot of feedback on the characters you draw. each point is marked with a circle for good points and a tick or triangle for bad points. Obviously it is all written in Japanese, but at the most basic level the graphical feedback is usually enough. The characters are also given a percentage score based on balance (out of 60) detail (out of 20) and line thickness (out of 20)
Personally I think Kakitorikun (the first edition) is a better title for learners. It covers the 1006 grade school kanji and teaches readings as well. The feedback on your kanji writing is fairly crude compared to Beautiful Kanji training however.
The kanken edition of kakitorikun has better feedback, like a cut down version of Beautiful Kanji, but overall it is far too complex if you don't already know the grade school kanji in my opinion. It's fine for the handwriting training but most of the reading and writing exercises will be frustratingly hard. I recommend the simpler first edition to start with it's much easier to get into.
Also none of these are games as such. They are more like textbooks with instant feedback.
kakitorikun site
http://100mas.jp/kakitorikun/
kakitorikun kanken edition site
http://100mas.jp/kakitori2/index.html
DS Beautiful Kanji training
http://www.nintendo.co.jp/ds/avmj/index.html
Site that might help you understand Beautiful Kanji training
http://xn--wgv71a119e.heilling.net/dsbi-moji-toreningu
I'm not sure what would help with dysgraphia.
I have noticed that second language writing is markedly different to writing in your primary language. (My Japanese handwriting looks crude, my wife's English handwriting shapes likewise look a bit crude but her kanji are fairly sophisticated.)
Using guidelines and writing in a larger size may help.
--edit--
afterthought
You don't have to write to learn kanji.
Reading and keyboarding are probably more used by second language learners than writing by hand. It might be better to sidestep the problem if you don't have an immediate need to write by hand.