[r] How to tell CRAN to install package dependencies automatically?

On your own system, try

install.packages("foo", dependencies=...)

with the dependencies= argument is documented as

dependencies: logical indicating to also install uninstalled packages
      which these packages depend on/link to/import/suggest (and so
      on recursively).  Not used if ‘repos = NULL’.  Can also be a
      character vector, a subset of ‘c("Depends", "Imports",
      "LinkingTo", "Suggests", "Enhances")’.

      Only supported if ‘lib’ is of length one (or missing), so it
      is unambiguous where to install the dependent packages.  If
      this is not the case it is ignored, with a warning.

      The default, ‘NA’, means ‘c("Depends", "Imports",
      "LinkingTo")’.

      ‘TRUE’ means (as from R 2.15.0) to use ‘c("Depends",
      "Imports", "LinkingTo", "Suggests")’ for ‘pkgs’ and
      ‘c("Depends", "Imports", "LinkingTo")’ for added
      dependencies: this installs all the packages needed to run
      ‘pkgs’, their examples, tests and vignettes (if the package
      author specified them correctly).

so you probably want a value TRUE.

In your package, list what is needed in Depends:, see the Writing R Extensions manual which is pretty clear on this.