Sleeper trains Amsterdam to Italy (Naples)

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I really want to experience traveling in a sleeper car. I'll want to get a private room. I will Depart Amsterdam May 13. When I look up the Trenitalia it seems very confusing. Where to by tickets, what is the price of the class ticket and the sleeper?
How many tickets will  I need? Will I have to change trains and then purchase additional tickets? On the Trenitalia site I saw more than 1 Thello train, old and new. What are the sleeper single cabins like? Hope to hear back from you. Thank you.

Welcome to reality.
There is no possiblity to cover your travel by single ticket. You must find optimal combination of tickets by your own and buy them from different sources.

It's shame to admit, but National Railways of each contry in Europe consider the necessary to maintain the possibility to sell direct tickets and provide support to wishers to travel with changew only to the nearest and medium-distant other countires, while support same possiblities to countries which are "too remote" is considered as "unimportant" and "not worth the efforts".

And Italy and Netherland - is one of those cases, when each railway threat other as "too remote" and suggest "normal people anyway don't travel so far by train".

Railway tariff which allow to cover whole way - simply doesn't exist.

Amsterdam to Naples - are really abit far from each other, distance between them is longer, than most people find acceptable to overcome by train.

It's not possible to cover this distance neither by direct overnight train (i.e. during evening-night-morning) nor by combination of daytime services (i.e. by start in early morning and finish before midnight).
You need at last 19 hours to overcome this distance and combine overnight train and daytime services.

You may travel by different ways (with different detours to West or to East) and night section can be placed on different parts of your travel (closer to Amsterdam, closer to Naples or wherever in the middle).
Basically you may choose between following 4 overnight sections. Also in each case I'll provide minimal price in case of single accomodtaion of early purchase (first minute price)/full price (price before departure), and by italics - minimal price of supplying daytime sections in 2nd class and place to buy

1.
Nightjet Duesseldof to Innsbruck 129/199 EUR without shower (only sink in compartment), -/219 EUR with private shower.
ICE Amsterdam - Duesseldorf 19,9 EUR (buy at bahn.de)
EC Innsbruck - Bologna from 24 EUR (but at oebb.at)
Bologna - Napoli alternatives:
Trenitalia from 35.9 EUR by high-speed train or from 9.9 EUR IC
or .italo Bologna - Naples high-speed train from 24.9 EUR

2. thello Paris - Milano
115/290 EUR with shower (it's temporary promo wich will last till midlle of may action, when you got cabin with shower on price of cabin without shower, that's the reason of such ratio, normally 115 is first minute price of cabin without shower, but not of a cabin with shower)
Thalys Amsterdam - Paris, 35 EUR
Milano - Naples alternatives
Trenitalia (39.9 high-speed or doable by intercities by 9.9*2 EUR) or .italo hispeed from 38.9 EUR

3.nightjet Munich - Rome
129/199 EUR without shower, -/219 EUR with private shower.

bahn.de Amsterdam - Munich, travel with changes (usually 1 change in Franfkurt is enough), from 49,9 EUR.
Rome - Naples alternatives
trenitalia
IC from 9,9 EUR
high speed trains from  16,9 EUR
local trains - fixed price 12.3 EUR available even before departure
or .italo - high speed trains from 9.9 EUR


4. trenitalia Alessandria - Naples -/202,8 EUR, only option without showers in this train

ICE Amsterdam - Basel from 39.9 EUR, buy from bahn.de
Basel - Domodossola from 23 CHF(spartageskarte from SBB.ch)
local trains Domodossola - Novarra -Allessandria 10.65 EUR, buy locally.

also keep in mind - most scenic section of the way - is crossing Alps.

If you go via Paris and Thello - you miss it.
If you go by overnight train from Munich and Rome - you miss it
If you go via Basel-Domodossola - in general you'll go via Alps whole Swiss section of your way, but the most significant mountain ridge on your way you overcome by 20 km Simplon tunnel, i.e. train doesn't really climb up.
But on other hand - with spartageskarte you may choose any itinerary via Switzerland (with any detour, including Bernina pass or whatever you like).
And only if you via Innsbruck - Bologna during daytime - you cross mountains "honestly" via Brenner pass, climbing to 1374 meters. That's quite serious argument for choosing this option.

Maxy already did a great job in explaining your options, but perhaps the only thing which makes sense to add here is that maybe you don't really need to travel all the way from Amsterdam to southern Italy by land, which is way too time consuming. Imho, in your situation it seems to make more sense to fly half way and then use an overnighter, in this case you can depart on the afternoon on May 13 and be in Naples already next morning. Most obvious choice would be to fly to Northern Italy (Turin or Milan or Genoa) and then take one of the direct Italian domestic overnight trains to Naples (most likely will look something like this). Or, if you want some adventure, you can fly to some place like Vienna and then take an overnight train from there. Or do a reverse, i.e. use an overnighter first and then fly to Naples.