adverb, noun which may take the genitive case particle 'no'
strongly (hoping or requesting); earnestly
Used with verbs of hoping or requesting, emphasizing a strong desire for something difficult to achieve. Often appears as たっての願い or たっての希望.
たっての願いで、彼に協力を頼んだ。
At his earnest request, I asked him for cooperation.
たっての希望で、その仕事を引き受けた。
I took on the job at her strong request.
強く is a general adverb meaning 'strongly' or 'intensely', while たって is limited to hoping or requesting and carries a nuance of earnest entreaty.
The derivation is uncertain. The kanji forms are ateji, and the word is usually written in kana.