The input image
The most critical part for a good result is the input image itself. Good contrast and a similar color distribution as the parts in lego art set of choice helps to produce good results. It is worth trying a few different images and different zooms/crops of your subject. A homogenous background can be beneficial. Therefore, you can replace the background with any plain color in an external image processing program.
Color adjustments
Play with the color sliders and see how it affects the resulting mosaic. Use the hue slider to adjust the overall color of your image and bring it closer to the available colors in the lego art set. Brightness and contrast can be used to shift the image into a similar brightness scale as the available colors. Sometimes it helps to desaturate the image to get better results. A less saturated image will favor brightness differences instead of actual color differences. If background takes up a lot of space in your input image, consider replacing it with black (RGB 0,0,0) in an external program. You can then use "Ignore black regions" to exclude these background regions from processing. The algorithm will only fill the object and can better distribute the available colors.
Choice of lego art sets
The resulting mosaic of course heavily depends on the choice of the lego art set that you have available. If you do not already own a lego art set which restricts your choice, feel free to experiment with different sets or even combinations thereof.
Part limits
I found the results to look more pleasing when the number of parts of a given color is limited through the choice of the brick sets. When part numbers are not limited, the result often looks flat with little color contrast. A restriction in the number of available parts forces the algorithm to use suboptimal colors, thereby, increasing the color space of the resulting mosaic. As a result, the mosaics will get a more artistic look. You can increase this effect by further restricting the part numbers through entering decimal numbers into the input field for the set count (e.g. Beatles: 0.9).
Multiple images
If you want to build multiple mosaics (and you own the necessary number of lego art sets to build them) it is advisable to produce a source image with all images stitched together, and adjust the mosaic size and brick set numbers accordingly. As a result, the algorithm is free to distribute the available parts between the source images.
Final real-world adjustments
Don't take the generated mosaic and instructions as the final truth. When you are done assembling your mosaic, you can still switch individual bricks to create a more aesthetically pleasing result. That's part of the fun.
Try other pages for lego art mosaics
There are two other great sites, that I know of, that allow you to generate lego art-based mosaics with part limitations. Feel free to try them out.
Lego Art Remix
LEGO Art Mosaic Generator