Where to place your own ldraw pars I just started designing LEGO instructions. I used LeoCAD to built the model and Web Lic to create the instructions. 

Being a Linux developer, Windows is hard to use and access.

It seems bricks should be created with MLCad, but it does not seem to run on Linux. Anyway, I tried creating a brick with vim and information in this tutorial, but can't find where to place my unofficial parts (.dat) files and how to see the result.

How to use my own (created by me) parts ?
Cannot comment on LeoCAD as I've never used that, but LDCad is far superior over MLCad and it works well on Linux natively. LDCad uses same part library format as MLCad - LDraw. You cand find LDRaw file documentation here.

It is not clear from your question if you wish to create your own parts or just add the ones not currently available in official LDRaw package. If you are interested in the later - there is a guide. It can be easily adapted to Linux.

In case your intention is to create your own parts LDraw contains a nice documentation explaining the file format. The process of adding a newly created part into the library is the same as explained in the section above. Just make sure to put your file for the new part into the /Unofficial/parts directory under your LDRaw package installation directory and make sure to run make-list to include the newly added files in the list of available bricks to build with.