"We do not like him."
Translation:My go nie lubimy.
First of all such a construction puts an emphasis on 'him' so you need to use 'jego' rather than 'go'.
Secondly, the meaning is different. 'Nie jego lubimy' = 'It is not him, who we like' = we like someone, but not him.
As a rule, 'nie' has to stand in front of the word it negates.
Similar as in the other sentence: Him we don't like (but we like her).
Would 'him' in this example be in genitive? And if so, isn't the genitive for 'him' 'niego' or something or am I mistaken?
Yes, it's Genitive (because of the negation), Accusative "Lubimy go" would look identically.
"niego" and other n- forms are 'special forms' of pronouns used after prepositions. "(Nie) idziemy do niego" would be "We (don't) go to him/to his place".