Your best bet would be to set an intent with type application/vnd.android.package-archive and then start installer activity:
Intent promptInstall = new Intent(Intent.ACTION_VIEW)
.setDataAndType(Uri.parse("file:///path/to/your.apk"),
"application/vnd.android.package-archive");
startActivity(promptInstall);
Check out Install Application programmatically on Android
Also, note, that for security reasons, you cannot install an apk 'silently' or automatically without user agreement. Therefore, the best you can do is like above.