How do I achieve this liquidy/melting effect in the image?

First make sure you are using a high res image.   This is something you should always start out with.  Something that has a good size and rougly 300dpi or so.  Remember after you create the effect, you can always scale it down. 

Plus this will help if you are creating posters, postcard invitations, etc. Or whatever.

Select the desired image.  Then under filters choose the Liquidfy Filter...

Top tool is what you want.... this is called the Forward Warp Tool.

Then adjust the Size, Pressure and Density.

I do not think it will ever look prefertly crisp.  But having a high res image that is good size will help you keep it look smoother.

Something you have to play with as without havnig the exact image I can not guess-timate which size, pressure and density you need to create what you want.  However, it is a fun tool to play around with and find out the solution.