How can I programmatically populate a PDF's pdfcreator field?

pdfTeX

The TeX compiler pdfTeX already sets some keys as default:

/Producer (pdfTeX-1.40.15)
/Creator (TeX)
/CreationDate (D:20140506215840+02'00')
/ModDate (D:20140506215840+02'00')
/Trapped /False
/PTEX.Fullbanner (This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) kpathsea version 6.2.0)

Some of these entries can be overwritten via \pdfinfo or via \hypersetup if hyperref is loaded. A nice feature of pdfTeX is, that if it detects the setting of standard keys in \pdfinfo, then pdfTeX suppresses its setting to avoid duplicate keys in the same dictionary with different values. Therefore it is possible for the user or hyperref to override the setting of pdfTeX.

The default of hyperref's values are (line breaks added for better readability):

/Author()
/Title()
/Subject()
/Creator(LaTeX with hyperref package)
/Producer(pdfTeX-1.40.15)
/Keywords()

Also the other entries can be set by hyperref (pdfcreationdate, pdfmoddate, ...)

Version number of pdfTeX

Since the version number scheme of pdfTeX has changed in the past, the algorithm for the versions specification is quite complex, from hpdftex.def:

\def\@pdfproducer{pdfTeX}%
\ifx\eTeXversion\@undefined
\else
  \ifx\eTeXversion\relax
  \else
    \ifnum\eTeXversion>0 %
      \def\@pdfproducer{pdfeTeX}%
    \fi
  \fi
\fi
\ifx\pdftexversion\@undefined
\else
  \ifnum\pdftexversion<100 %
    \edef\@pdfproducer{%
      \@pdfproducer
      \the\pdftexversion.\pdftexrevision
    }%
  \else
    \ifnum\pdftexversion<130 %
      \edef\@pdfproducer{%
        \@pdfproducer-%
        \expandafter\@car\the\pdftexversion\@empty\@nil.%
        \expandafter\@cdr\the\pdftexversion\@empty\@nil
        \pdftexrevision
      }%
    \else
      \ifnum\pdftexversion<140 %
      \else
        \def\@pdfproducer{pdfTeX}%
      \fi
      \edef\@pdfproducer{%
        \@pdfproducer-%
        \expandafter\@car\the\pdftexversion\@empty\@nil.%
        \expandafter\@cdr\the\pdftexversion\@empty\@nil.%
        \pdftexrevision
      }%
    \fi
  \fi
\fi

Other drivers

Package hyperref also sets information entries with the other drives. But it can happen, that the driver insists of its setting for some of the keys (/Creator, ...).

\pdftexbanner (introduced in pdfTeX 1.20a; see the pdfTeX user manual) produces the output:

This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/W32TeX) kpathsea version 6.1.1

and contains the version of pdfTeX being used to process the document. You can extract just the pdfTex version using the following:

\makeatletter
\def\extractpdftexversion#1-#2-#3 #4\@nil{#3}
\edef\pdftexversion{\expandafter\extractpdftexversion\pdftexbanner\@nil}
\makeatother

Here \extractpdftexversion expects an argument text of the form <stuffA>-<stuffB>-<stuffC> <stuffD>\@nil, and only returns <stuffC> (that is, the text delimited on the left by - and on the right by a space ). You can add it to your PDF document properties using

\pdfinfo{
  /Producer (pdfTeX \pdftexversion)
}

\documentclass{article}
\makeatletter
\def\extractpdftexversion#1-#2-#3 #4\@nil{#3}
\edef\pdftexversion{\expandafter\extractpdftexversion\pdftexbanner\@nil}
\makeatother
\pdfinfo{
  /Producer (pdfTeX \pdftexversion)
}
\begin{document}
\pdftexbanner
\end{document}

No explicit need to use hyperref (see Is hyperref really the best way to add metadata to a TeX file?).