The Problem

There are many webpages that try to tell you how to typeset polytonic (ancient) Greek using LaTeX. But most are out of date.

The simple way on modern Linux or MacOSX systems is to use XeLaTeX and the fontenc package.

What you need

I'm assuming you already have an editor that can cope with Unicode. All you need, then are these packages:
      texlive-xetex
      lmodern
      ttf-freefont
    
Then your document preamble should look like this:
      \documentclass[a4paper] {article}
      \usepackage{fontspec}
      \setromanfont{FreeSerif}
    
The \usepackage{fontspec} line means that XeLaTeX will honour the UTF-8 font encoding, and switch encodings appropriately.

fontspec can be used to do lots more, see its manual for details.

\setromanfont{FreeSerif} chooses a standard font for the text. You'll need a font (such as the Gnu FreeSerif font) that includes the glyphs for polytonic Greek.

Then in your text, you can freely intermingle Greek and English, thus:

	Ἐν ἀρχῇ ἦν ὁ λόγος,\\
	In [the] beginning was the word,\\
      

Build using xelatex

	xelatex mydoc.tex
      
and the result will be put into mydoc.pdf

A complete example

      \documentclass[a4paper,12pt] {article}
      \usepackage{fontspec}
      \setromanfont{FreeSerif}
      \usepackage{geometry}
      \begin{document}
      \section*{John 1:1}
	Ἐν ἀρχῇ ἦν ὁ λόγος,\\
	In [the] beginning was the word,\\
      \end{document}
    

Run through xelatex, and the result is:

Jn 1.1 in greek and english

Acknowledgements

Based on http://existentialtype.net/2008/07/12/fonts-in-latex-part-one-xelatex/