@コッド Your example should be "Nihongo ga hanaseru you ni narimashita". Also, I think the particle would be "de".
@Hnrutt The dictionary form of a verb is identified by ending in ~u or ~ru and marks the informal present affirmative of that verb.
The only way I can see "hanaseru" being considered in the "dictionary form" is by noting, that other dictionary form verbs can be formed from dictionary form verbs by using certain endings to give certain meanings.
So while "hanasu" is dictionary and means "(I) speak", "hanaseru" can be considered the dictionary form of the verb "to be able to speak", though these won't actually appear in the dictionary.
Basically, what I'm saying is that "hanaseru" isn't just a certain conjugation of "hanasu", it's actually a new verb and can be conjugated just a normal verb like "taberu".
So you could say "hanasetai" which means "I want to be able to speak". In this case, you could start from hanasu -> hanase -> hanaseru -> hanase -> hanasetai, but instead you already have the verb for "to be able to speak", so you can just add the ~tai ending.
Also, this would explain why in the polite form, "hanashimasu" means "I speak" while "hanasemasu" means "I can speak". The ~masu form is actually conjugated from the new verb "hanaseru". Basically, I think it might be considered being tentatively in it's dictionary form because it can conjugate like any other verb.
(If this is entirely wrong, someone tell me; I've never really thought much about potential verbs being separate and conjugating normally)
Summary: Dictionary form ends in ~u/~ru. By using the stem (or 4th base) of a normal verb, the ending ~ru can be added. However, this effectively makes a new verb, which can be conjugated to the same extent that the original verb, and can be considered the dictionary form of "to be able to X".