(A) ように～てほしい is an expression in Japanese which combines the auxiliary verb, ようだ, with the てほしい structure (indicating that you/someone else wants someone to do something). As ようだ is being linked to a verb which comes later in the sentence, it will take on the adverbial form ように.
This expression is used to highlight that you (or someone else) want someone to do something ‘in the way of (A)’.

• 弁護士（べんごし）になれるように頑張（がんば）ってほしい。
  I want you to work your hardest, so that you can become a lawyer.
• このチームが大会（たいかい）で勝（か）てるようになってほしい。
  I want this team to be able to win a tournament.
• お前（まえ）にはスポーツ選手（せんしゅ）のように走（はし）ってほしい。
  I want you to run like an athlete.
• もっとアナウンサーのように話（はな）してほしい。
  I want you to talk more like an announcer.

Note that when ように follows a noun, の will be required.
Either に or が may mark the thing that the speaker ‘wants’ to do. It is more common for に to be used to mark people, while が will often mark events/objects.

• あなたに漢字（かんじ）を書（か）けるようになってほしい。
  I want you to be able to write kanji.
• 事故（じこ）が無（な）くなるように運転（うんてん）をしてほしい。
  I want people to drive in a way that will reduce accidents.