"He does not feel well."
Translation:Ele não se sente bem.
Because "bom" is "good" and "well" is "bem". One can be "good", but cannot be "well". Rather, one can do well in something, but cannot do good in something.
Ele é um bom aluno de Português = He is a good Portuguese student
Ele foi bem na prova de Português = He did well in the Portuguese text
The verb is being used reflexively here, all that's different is the placement of the reflexive pronoun "se". In Brazil it is more common to find the pronoun before the verb. You mention senti-se, but you probably meant "sente-se"; see the comment by joaopsousa for the reason why "sente-se" is not used in sentences like this even in Portugal.
Portuguese uses the reflexive form to distinguish the idea of feeling something within from feeling something with the senses. The general idea of reflexive verbs and how to use the reflexive pronouns is briefly described here:
"Se", in this case, is the reflexive pronoun for "Ele/Ela".
In "Ele não se sente bem", he is clearly talking about himself. In "Ele não sente bem", it's not clear what he doesn't feel well. I know that in English it's pretty clear he is talking about himself. However, in Portuguese it's not (mostly because "sentir" has multiple meanings and because "well" is placed differently in a sentence than "bem"). So when it's written "Ele não sente bem", we don't know what he doesn't feel well.
"Ele não sente bem", can be a phrase of a larger sentence like "Ele não sente bem a mão dela" [He doesn't feel her hand well -- I'm not even sure if "feel" is the best word to convey the meaning here] or "Ele não sente bem o cheiro de comida" [He doesn't feel the smell of food well -- dito; maybe "sense" is better here].