Oi, Júlia. "Would like" é simplesmente "gostaria". Ex.: I would like to register and start earning money immediately. Gostaria de me registar e começar a ganhar dinheiro imediatamente. "Feel like" pode ter duas ideias, a primeira é quando ele é seguido de um substantivo. E significa "Eu me sinto como..." Ex.: I feel like a little girl. Eu me sinto como uma menininha. A segunda ideia de "feel like" ocorre quando essa "collocation" é seguida de um verbo. E dá a ideia de "ter vontade", "querer" Ex.: I feel like dancing. Estou afim de dançar. É isso. :)

Would like is a ver in conditional form. However, feel like is an phrasal verb which express you want to do something usually at the moment of speaking.

Gostar de Would like (+ noun or infinitive of verb -- for example to travel) Ter vontade de Feel like ( + noun or gerund -- traveling)

-'Would like" it to say that you desire somthing, for example i would like to be rich/ i would like to know others countries/i would like to a better job -"feel like" it to compare you with others people or things,i for example i feel like a baby when i eat icecream/ i feel like a stupid when she tell me that/ i feel like a computer because i have many knowledge