Can I travel by boat from the North Sea to the Mediterranean passing via Paris?

There are canals and rivers suitable for boats that connect the both seas.
And while rare, it is not exceptional to travel the distance. It takes time and quite a bit of money, preparations and patience.

You mention wanting to start in the Netherlands. That means that you will need to pass through Belgium and/or Germany and will have to consider three countries rules and regulations.

If you own a boat it has to be suitable, big enough to cross and where needed use the rivers and busy canals, small enough to fit in the locks and under the bridges you will meet on the way.
Most of the canals in Belgium and in France will restrict your dept as well as width, so if you own a sharp yacht it might be unsuitable. And the same if you own one of the real big motor yachts. And you will also need to consider low bridges and even tunnels. The motor needs to be in a good condition and you need to have all needed paperwork and insurances for such a long travel.
It is likely you need a license for using the boat on the waters in the countries involved, certainly if it is over a certain length or has more than a given power, (motor power in ratio to weight of the ship and so on.)

If you need to rent a boat you will have to find a rental company happy for you to take a boat out that far. And it will not be cheap, boats do rent out at many rates but cheap is mostly not one.
One way rentals are even less likely.

There are many sites in Dutch about this kind of travel.
Picaro
ANWB for France pdf
ANWB for Belgium
I could not find English language sites at first and did not look for French ones as I do not speak that language.
But that same Picaro site has a French option.
An English language site with links points out the three basic routes, one of which is around France rather than through it.

Map of waterways in France, external link to the site of www.michaelbriant.com

For general routes in France, see this map, from a link in the comments to the question by A.E
This is a site with links to maps in Flanders (Dutch speaking part of Belgium, site in Dutch)
More maps with usable canals in Belgium in this aswer.

Source (not alive anymore) http://yachtchartersneek.nl/waterkaart_Belgie_vaarwegen.htm

Online maps and sites with information are good for planning, when on the water you need charts for the area, best with the almanac/guide with the information on locks, bridges and all the other details you will need to cross the area. Likely one of the ANWB sites will have links, if not the one I linked to above, then one you can find by using google. Or just search online for those charts.

It is possible to include Paris in your trip, but Paris is not a good city for visiting boats. There are a few marinas, but those are likely to be rather full with boats owned by Parisians but at times visitors can find places.

One of the canals in Paris has plenty of tunnels. Photo by Willeke and can be used freely.

This canal is off the Seine and as far as I understand it is not on the likely routes for yachts for cross country travels.

Count on about 6 km/h, add an hour per lock, can be a day if you are unlucky, and you will see that this tour is something for retired people.
I guess it will be 250 hours of actual 'sailing' but you will not be able to do more than 8 hours on most days, and might be stopped after fewer hours by a lock not working if unlucky.
And that is one way, (but calculated from the center of the Netherlands, you will save a few hours if you start farther south.)
(And my guess is based on the total distance by Google maps, which used a different route, so it really is a guess but confirmed by a friend who owns a boat with which the family has been to Paris.)

If you are an experienced boat owner, you will know that you need to know a bit about emergency repairs, how to react when things go wrong on a busy waterway or when you are in the way of a potential dangerous bigger ship.
If you are not experienced, get a basic course, aimed at the boat you will use, general emergency information, how to communicate with bridges and locks, how and when to pay your fees when you overnight in a marina and how to handle water intake and waste.