It's not possible in my opinion and experience.

There's nothing that you can do. They see it when they're willing to be honest with themselves. Some of them never even get there. The biological tie that they have to their kids is all that they need to live in constant denial. It's disgusting.

There's not much you can do.  I've been with DH for 7 years and he still doesn't like me saying anything, even when he asks for my opinion.  So I'm careful to not give an opinion (although one slips out every so often).  He has to do everything himself to see the issue.  I don't parent, I don't try to parent.  If something bothers me I leave the room or the house.  He deals with it, not me.  That's the only way he's ever improved on anything.  It's never, ever been MY saying anything or talking to him.  Even if I say something good about a book I read, he'll downplay it or dismiss it...then he reads about it, years later, and he thinks it's amazing!  My response:  "Oh, interesting."

Do you have friends with very good or very bad kids? Spend some time with both. Talk about how amazing the good kids are and about how the parents did a great job. After hanging out with the bad ones wait until your SO says something about them and then slide in a "other people's kids are always more annoying than your own."

You could be blocking the view. Don't overcompensate, overfunction, facilitate, assist, enable, or help in any way with your SO's kid(s). Dont nag, vent, complain, demand, or push. It only makes you the bad guy, and muddies the waters. Instead, let your SO feel the consequences of their poor parenting. 

I tried for years to get my DH to engage more with his kids, and to warn him that his eldest was behaving badly and needed correction. It fell on deaf ears. Finally, I disengaged from the crazy. Once I wasn't involved anymore, my DH had to deal directly with OSD, and got a full blast of her narcissism and eagerness to use us for personal gain. What I'd been complaining about for years suddenly was affecting him, and he didn't like it at all. 

I went back and skimmed your earlier posts.. and in a nutshell, it appears you have two rather young (4 and 8 ) children who have had to deal with the "loss of their mother" due to issues with drugs... which means they likely had a very chaotic household even when they were with her.. and they have had to adjust to a new home... loss of mom in their lives and a new "mother figure".   Even the split up of their parents too.. assuming their parents were together not too far in the past.  All at an age where they don't have the emotional tools to truly deal with all this change and grief.

So, of course they are acting out to an extent and in some ways you are a reflection of the biggest change.. the loss of their mother.. and the competition of sharing dad with another person.  In a way, it's not personal at all.. it would be the same for anyone in your position.  They don't hate you.. but they are having a hard time dealing with what you represent.

NOw, kids aren't necessarily subtle.. they feel hurt.. so they will act in ways that hurt others... and they have not been parented well.. by EITHER parent.  I would lay bets that your SO has an expectation that you mother these kids.. that you act in a parental role.. including discipline and childcare.  So you probably have expectations that are over and above what these kids can do right now.. and that frustrates you.. they sense the resentment and it circles around to more hurt and bad behavior.

Then, when you complain to him that his kids are being brats.. he takes that as a personal attack on him and his children.. he defenses up.. he stops listening and you get further frustrated because he can't "see how horrid they are".

So... how do you deal with this and improve the situation?

First, I like the concept of by trying to prove yourself with the kids... taking care of them... "playing house" so to speak.. you are allowing your SO to avoid dealing with his kids in enough situations to see where they have holes in their development.  Perhaps if he was the one who had to clean all the messes.. he would be more inclined to prevent them?  Maybe if he dealt with the bratty turn up of noses at dinner, he would teach them better manners and helpthem become better at trying new foods?  Maybe if he were the one to enforce bedtime, he would deal with the backtalk and tantrums and work on a better standard of behavior?

Second, there is always the possibility his kids might benefit from some counseling.  You can approach this from a place of concern.  "Honey, I think the kids are struggling with the situation.. loss of mom in the picture and all the changes.. do you think it might help if they got some counseling to help them adjust?  I think we would find they would be happier"

Third,  There is disengagement (part of the first part.. let him BE their parent).  Disengagement isn't just stepping back physically.. but ALSO emotionally.  You have to get to the point where you "don't care what people do that don't care about you" to some extent.  They roll their eyes?  no reaction.. just ignore.  They are kids and by DEFINITION.. immature..  And.. again, it's really not meant as personally as it appears.. even though I am sure it feels like it.. it really is probably less about you than you think.  So.. let him parent.. interact with the kids only to the extent that they initiate it in a positive way.. train them that the only reaction they will get from you is when they are behaving well... they are in their brattiness.. nautiness.. then you simply leave them to their father to deal with.

Fourth is picking your battles.  Surely there is a short list of non-negotiables when it comes to their behavior.. physically destroying your things?  Leaving mess in the living room.. address those first.. make it your SO's responsibility to make those things stop... either by parenting or exclusion from areas of the home.. whatever HE has to do to make that happen.

As long as you point it out, you will always be the bad guy. You need to step back and let him deal with things all on his own.  Don't ever say "your child did this" say "the dishes in the kitchen sink need cleaned" etc.

And here's the deal - even when he realizes his child has issues, he will still love her and she will still be around. Even if she has done something to hurt you. 

Just another reason step-anything is so unnatural. 

Sometimes not hearing it from you but from someone or something else may help. Sometimes talking to our SOs about these topics ad nauseum sets the stage for failure in future conversations. Pride kicks in. Emotions are in overdrive. Right fighters take over. Someone maximizes while the other minimizes. 

A friend of mine told her DH about "Someone's" kid who was cursing out their stepparent every time the Dad left the home. He of course was horrified that "someone's" child would do such a thing. It was a good thing SD wasnt like that... She then showed him a video from her phone. It did solve the issue. 

I would imagine this would only work for a specific issue. 

Everyone else gave great advice.  I only want to add that you can opt out of any contact with the kids.  You don't have to go out to eat with them, vaction with them, etc.  You also don't have to babysit them.  Avoid them whenever possible.

Unless you need evidence for your own purposes. I ended up with video of a skid doing exactly what I said and DH was denying and he still made excuses for the skid.