Please describe your work flow..............as in what have you been doing and show us your work.
There are 206 frames in this animation. You start by selecting both frame 1 and layer 1 and remove the white background ( I used the Magic Eraser Tool). Then you do the same for frame 2 layer 2................and so on.
Here are the first 30 frames with the white BG removed. I have added a blue background.
The white specs you see were part of the original substrate of the female character.
If you use the Quick Selection Tool to make a selection, then hitting delete.
You are still going to have white specs in the outline because of the selection process (Quick Selection Tool). The small white specs on the border of the character will be included in the selection and will show up with any contrasting background color.
If you use the Magic Wand Tool set to 0 tolerance to make the selection and then hitting delete, you can get a fairly decent result.