Dear Leo1977,

First welcome to PC!
As the world is today I find many children suffer as a consequence to unfortunately primal parenting. This child is not owned by one or the other.
They are both in it for the next 20 years...

Here is the truth from the boys perspective, he wants his parents to get along and to be together again in a loving relationship. (That's not going to happen) So the child is torn.
At age 4 he lacks the verbal skills to communicate how he feels.

The father's kid hates me and when the kid speaks of me he gets very angry so the kid started telling the truth to avoid conflict between his father and me.
We are worried that the situation has led the kid to use lies in order to avoid conflict.
????

If you are good to his mom and him I'm sure the child doesn't HATE YOU.
He is repeating what the bio-dad is expressing to please his dad, so don't take it personally. And I'm 70% sure there have been negative talk on the weekend about mom, which the child is holding inside. Therefore you have a rational right to be concerned.

I think you need to make sure how much you are committed to the relationship.
For now when the child is around you can relate boyfriend/girlfriend status but keep the romance for the weekends.
Simple explanations are best, telling the truth is the right thing to do and people get hurt when you lie to them.
Allow him to feel whatever he does and your validation of his emotions can go hand in hand of times when you felt sad, unheard, didn't get what you wanted, or angry.
Just use age appropriate language and be kind.