I would suggest making this on the stove top, or in the microwave, instead of in the oven.
What is happening is that the oil from the cheese separates from the rest of the dish.
If you are using bottled salad dressing and cream cheese, there is already many thickeners/stabilizers in those.
(Xanthan Gum, Guar Gum, Carob/Locust Bean gum) I suggest that you slowly melt the cheese, along with the cream cheese & salad dressing on low heat .
An immersion blender would help to make sure the lumps are gone, after adding the hot sauce, but before you add the chicken (as long as you want to have the chicken texture).
From my experience, the oily separation with the cheese is caused by it over cooking.
If you aren't a cheese snob (I am), you could substitute Velveta cheese for the cheddar/jack cheese.
Do you remember their commercials with the oily gooey messes with cheddar vs Velveta?
It is the same principle at work here.
cooking.stackexchange.com/users/15303/Kristalyn Mowers
Although baking is a totally adequate way of making this dish quickly.
If you want to improve presentation, for parties and so on, I would recommend switching to making it in a crock pot or something like that (keeps it warm, contained).
If you wanted to add a stabilizer at this point, I would recommend agar-agar (boil in broth, fold into dip - use a smallish amount so that it works as a thickener and stabilizer but not like gelatin).
As for a component stabilizer to just add to the recipe, all of the ones listed in Aaronut's comment and Kristalyn's answer would work.
My preference, especially with a recipe like this would be to use mustard, prepared or ground.
cooking.stackexchange.com/users/15/mfg
Just made some buff chic dip in the microwave and it did NOT get greasy.
Even after sitting out for awhile.
What I did was microwave the cream cheese and a couple good squeezes of Ranch for 2 minutes.
The cream cheese was stil in block form after microwaving (not melted) but mixed together real easily.
Don't over cook the cheese!!!!
Then add chicken and buffalo sauce to taste and microwave in 30 second increments until hot.
I'd show you a picture if I could.
It looks creamy and delicious.
:)
cooking.stackexchange.com/users/30930/Sara
+1 to mfg for the crock pot suggestion.
That is how we make it.
To expand a bit, we do not put the cheese on top.
We find that it hardens into a skin once it cools a bit, making it very difficult to dip anything into it.
Instead we combine the shredded cheese with all the other ingredients at once in the crock pot.
We do cook the chicken separately, boiled usually.
The key to a consistent texture there is to finely shred the chicken in a food processor rather than fork or hand shredding, as larger or longer pieces of chicken make for a lot of broken chips.
We actually just made a batch of this yesterday using boneless skinless thighs rather than breasts and even with the higher fat content we had no problems with separation or oil run off.
I also want to add that we have tried putting larger chicken pieces in the dip and then using an immersion blender to smooth out the whole mix.
I don't recommend it.
The chicken doesn't end up uniform in size, and when hot the dip combines easily with minimal stirring so you don't gain anything by such rigorous blending.
Late edit: I want to add one more thing about doing this in a crock pot: don't let it sit too long on its own with the crock pot on.
It will burn.
Even on the lowest setting, the parts that are touching the sides will scorch.
If it happens a little bit, you can stir it in and it's not so bad, but it's definitely not at its best that way.
This dip reheats really well either in the crock pot or in a microwave, so don't be afraid to let it cool and reheat it.
cooking.stackexchange.com/users/29207/briantist
When I make it, I purposefully cook it to the point of oil separation, to get the taste and texture that I (and my audience) prefer.
Otherwise, it has an overly-creaminess, and less pungence.
Again, that’s just us.
I frequently serve it without achieving separation if I'm in a hurry, but look forward to reheating.
cooking.stackexchange.com/users/54971/Belsheen
I frequently see buffalo chicken dip showing up at parties. The idea of it sounds pretty good, but more often it ends up all separated with pools of grease like this:  (Source: browneyedbaker.com)  What could be done to stabilize the dip so it remains creamy and unseparated. I had the idea of adding egg yolk, but I suppose you'd have to at least be careful with it to prevent the egg from curdling.
How could buffalo chicken dip be stabilized?
cooking.stackexchange.com
<texture><emulsion><dip>
cooking.stackexchange.com/questions/30120
cooking.stackexchange.com/users/4489/Ray
Making this in the microwave or on the stove should give better results, being careful not to overcook to avoid separation of the cheese. You can try microwaving or cooking on the stove top. Try not to overcook as this causes separation. You can also add a stabilizer.
Try cooking in the microwave or on the stove top. Separation is caused by overcooking. An immersion blender will mix it very well. You can add a stabilizer like agar-agar.