Hopefully a pretty straightforward question:
How long does it take to get from Gare du Nord to Versailles by public transport?
Quickest method would be best, and I'd assume that would be train but I'm not ruling out buses, etc.
RATP trip planner gives 53mn at 10 o'clock, november 28th, 2013.
The route is:
Versailles-Chantiers is about 1km south-east from Château de Versailles.
There are two main stations in Versaille, Versailles - Château - Rive-Gauche on the RER C, and Versailles - Rive Droite on regular suburban network. The Château stop is about a 5 minute walk to the entrance, which the Rive Droite one is closer to 10. There is also Versailles - Chantiers on the RER C and local lines, but that's the furthest.
I'd suggest you use this Rome2Rio query for a general idea, which should weight up the different walking options from the different stations. You're basically looking at about 50-60 minutes, depending on connections. Be aware that a regular Metro/RER ticket won't be valid as far out as Versailles, so you'll need to visit a ticket machine and buy a ticket for the journey - you can't just use a regular carnet one for example.
You can also use the RATP trip planner to check exact times, but you'll need to search for each of the 3 stations individually, then factor in the walking times by hand. You'll probably either want to take the RER B to Saint-Michel Notre-Dame and change to the RER C, or the RER E to Gare Saint-Lazare and change onto a local SNCF service to Versailles-Rive Droite. (Metro 4 to Montparnasse then local train to Versailles Chantiers suffers from poorer frequency, but is a tiny bit faster if it works out)