I do tend to agree that this is probably best done in the original artwork - eg: Photoshop or Illustrator. But here is a simple solution in AE:
Duplicate your lines layer.
Top layer: Effect > Roughen edges : tweak to taste, but don't worry about any parts disappearing. Note you can set the sacle value for Roughen Edges right up to 1,000% which will give you a smoother / larger scale variation.
Set layer to multiply or darken blending mode if needed.
Second layer: Effect > Simple choker: set choke matte to reduce the thickness of the lines to the minimum you want to see.
The combination of the two layers should give you what you want.
Pre-compose when done.