Step 1:

Open the design you want to give a halftone dot effect.

Step 2:

Create a layer under the original design.
Create a shape that you want to convert into dots.
I copied the outside shape of my design and filled it with black.

Step 3:

You have to enlarge the shape so far that the later created halftone dots are visible below your original design.
My design is round so i can use the scaling option in Illustrator.
If your design has a special shape you have to manually adjust it.

Step 4:

Select Effect > Blur > Gaussian Blur

Use a Radius you want.

The result looks like this.

Step 5:

Select Object > Rasterize…

The resolution should be 600 dpi or more.

Step 6:

Select the object and select Effect > Pixelate > Color Halftone…

The color halftone dialog box will open where you can enter some numbers to adjust the appearance of the effect.
Set a number for the Max. Radius.
This will determine the size of the largest dot in the effect.
The screen angles should be equal.

Step 7:

We need to convert the raster image to vector.
Select the shape and select Object > Live Trace > Make and Expand from the menu.

The shape is now a full vector object.

Step 8:

Select with the Magic wand tool the white area and press delete.

Step 9:

Step 10:

Now we are able to assign the color and set the appearance of the design to overlay with the transparency menu.