How do I make my logo/mascot into a custom LEGO model?
The cheapest way to go about doing this is to design it yourself, like you said, using one of the LEGO cad softwares (LEGO Digital Designer, LDraw, MLCAD, etc.).  Then, buy the pieces and build it.  However, that takes a lot of time, and unless you are experienced, may not turn out so good.

The easiest (but also likely expensive) way to do this is to hire a LEGO certified professional.  The LEGO certified professionals are LEGO enthusiasts who have turned that passion into a business, and are officially recognized by LEGO for their work and skill.

http://aboutus.lego.com/en-us/lego-group/programs-and-visits/lego-certified-professionals

I would contact a certified professional and see how much they need to design and build your logo.