I didn't tell you because I didn't want you to know about it.
This would be the most natural way to say it.
I didn't want you knowing about it.
This is also acceptable.
In the first case, the direct object of want is "to know", which is an infinitive noun.
In the second case, the direct object is "knowing", which is a gerund noun.
I didn't tell you because I didn't want that you'll know about it.
This is incorrect, because "will know" (expanded from "you'll") does not form a noun to serve as a direct object; it is simply a future tense verb.
Edit: I previously reasoned that "do/did/does" was used as a helping verb and therefore the present tense of "know" or the continuous present tense of "knowing" would have to be used. However, the present tense verb paired to "did" in this case is want, not know.
@Cardinal raised an excellent follow-up in a comment asking about the validity of "I did not want you would know about it." I can say that is definitely wrong because "you would know about it" doesn't contain a direct object noun for the verb want. However, the sentence "I did not think you would know about it" is correct, because "you would know about it" itself forms a noun phrase as the direct object of think.
I'm a bit torn on exactly why "you would know about it" is a valid noun phrase for think and an invalid noun phrase for want, aside from it sounding completely wrong. The reason is almost certainly some intrinsic quality of the verbs themselves, but I'm not sure. Could use some help, there.