Inan upcoming campaign I will be playing a Lawful Neutral(escaped torture for stealing by making a pact and is has very strong opinions about honour) Duergar(MToF) Warlock of Juiblex(he was desperate) His name is Darrak Doomfist and I am wondering how I can deal with NPCs possibly being racist to my character without straight up murdering the NPCs. 

Any tips are greatly appreciated. 

The first time you try to murder an NPC then you are likely to be arrested/killed by the local authorities.

Having the above in mind might temper your character's actions.

A LN character is unlikely to keep working for a CE deity for long - or at least won't stay LN for long.

When considering how to have your character act there are two important considerations. In order of priority they are:

1) What would be a good contribution to the game and the fun of the other gamers around the table. 

2) What is the internal psychology of my character, given the back story and personality that I would find fun to play?

So the first thing you have to think or ask about is what kind of campaign your DM is running and what kind of characters the other players are making. See if you can think of ways that your character can complement or riff off of them. 

Then you can think about what kind of person you want to play and how they would personally approach racism directed at them. There are probably ways you can have your character react that are not disruptive to a game. 

One idea: If you want to play a psychologically conflicted character who is actually ashamed of being a duergar, maybe they hide that fact with a disguise or a hood or something?

Quote from Ophidimancer >>

When considering how to have your character act there are two important considerations. In order of priority they are:

1) What would be a good contribution to the game and the fun of the other gamers around the table. 

2) What is the internal psychology of my character, given the back story and personality that I would find fun to play?

So the first thing you have to think or ask about is what kind of campaign your DM is running and what kind of characters the other players are making. See if you can think of ways that your character can complement or riff off of them. 

Then you can think about what kind of person you want to play and how they would personally approach racism directed at them. There are probably ways you can have your character react that are not disruptive to a game. 

One idea: If you want to play a psychologically conflicted character who is actually ashamed of being a duergar, maybe they hide that fact with a disguise or a hood or something?

Quote from Farling >>

The first time you try to murder an NPC then you are likely to be arrested/killed by the local authorities.

Having the above in mind might temper your character's actions.

A LN character is unlikely to keep working for a CE deity for long - or at least won't stay LN for long.

 Thanks these really help!