The Tour de Croy was completed in 1765, according to the Chatillon site page Histoire de la ville. Construction took place in stages, starting in 1763. The height was eventually 50 feet in total.
It seems to have functioned as an observatory from the start, and it is referred to as la Tour de M. le Duc de Croy on page 61 of Histoire de l'Academie royale des sciences (1772) in an account by M. Le Monnier SUR UNE ECLIPSE HORIZONTALE DE LA LUNE, VUE A CHATILLON on the 23rd of December, 1768.
The Duc de Croy who built the observatory / tower was Emmanuel de Croÿ-Solre (1718-84), who was Maréchal de France (Marshal of France, the highest military honour in France). The Duc
manifeste un interet passionne pour les sciences de la nature. Il
construit une toure-observatoire a Chatillon...
Translation: had a passionate interest in the natural sciences. He
built a tower-observatory in Chatillon...
Anne-Emmanuel de Croÿ-Solre (1718-1784). Image source.
The observatory is mentioned several times in his journal and was used by the French astronomer Pierre Charles Le Monnier, among others. It appears to have cost the Duc 28,800 livres (pounds), a sum which was
bien cher pour une pareille folie, mais pas trop pour la réussite et
la beauté de' l'endroit, qui était généralement admiré, quand on y
allait par un beau temps, car cette vue écrasait toutes celles des
environs de Paris et maisons du Roi, et l'endroit fut fort utile aux astronomes;
Translation: dear [expensive] for such a folly, but not too much for the success and beauty of the place, which was generally admired, when one went there in good weather, for that view surpassed all those around Paris and the King's houses, and it proved very useful for astronomers;
According to a web page on the Tour Biret, the Tour de Croy was
de cinquante pieds de haut sur la butte de la glacière de M. Raffard.
Translation: fifty feet high on the mound of Mr. Raffard's cooler.
Unfortunately, I have been unable to find any drawings or plans of the Tour de Croy but a few of the features are mentioned in the Duc de Croy's journal. He mentions there was a belvedere (a summer house or open-sided gallery, typically at rooftop level, commanding a fine view). It was evidently built in stages; in Sept 1763,
Je décidai de l'élever de dix pieds de plus, au moyen de quoi
j'espérais dominer sur tous les arbres,...
Translation: I decided to raise it by ten feet more, by which I hoped to dominate all the trees,...
He also mentions the addition of two staircases and a balcony, and twice describes it as "charmant" (charming).
Emmanuel de Croÿ-Solre was also the driving force behind another Tour de Croy, begun in 1757 and completed in 1758. This tower, on the coast at Wimereux, had a defensive purpose during the Seven Years War.