In CS particularly, there is usually a step during the paper submission process in which you are required to disclose whether a part of your paper has been published elsewhere (or some similar phrasing). There you list the paper(s) your current paper expands.

Further, you state in the cover letter the circumstances in which your new paper was created (e.g. "in the conference proceedings, we show that [...], this approach is improved in this paper by [...], yielding [...] results. for better comparison, a new experimental setup was used, so the results of the conference paper are rerun in the new environment, which affected the conclusions [...]"  - you get the idea). In other words, you state in a few paragraphs (keep it short though, 1-2 paragraphs should be enough) what's new in you paper compared with the previously published one and how is that significant.

As for your paper, you use your conference paper as any other reference (including the related work review). The style depends, you can be personal or not, it doesn't really matter. I stick to the same style I use for other references, i.e. I don't explicitly state that the cited paper is written by me.