Il mio italiano non è buono. Non parlo un buon italiano. Non parlo bene italiano.

/bene/ is an adverb (near a verb) /buono/ is an adjective. In this case /to be/ is usded to join the subject to the adjective, that is why yiu need an adjective. This special use of the verb 'to be' is quite frequent! Note with another verb: Parlo bene l'italiano (adverb) Parlo un buon italiano (adjective) Capisco bene l'italiano (adverb) Capisco soltanto il buon italiano (adjective) Capisco l'italiano, ma se scandito bene (adverb) /essere/ has its special rules!

"Il mio italiano non è bene" it's wrong. The correct form is "Il mio italiano non è buono".

Bene(Ital.)=bien(Span.)=bem(Port.)=bien(French)=bé(Cat.)=Well (Adverb) Buono(Ital.)=bueno(Span)=bom(Port)=bon(French)=bo(Cat.)=Good (Adjective) As you can see it's similar in most Romance langauges. In English you also usually say "my Italian isn't very good" rather than "my Italian isn't very well". So it shouldn't be very difficult.

"buono" is an adjective so it refers to the noun "Italiano" instead "bene" is an adverb so it refers to verbs