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I have seen a postcard with a quote "You can achieve a lot with hate, but even more with love", signed "Shakespeare", in a French bookshop.

The quote is not exact, because I have translated it back from French. The original is “On peut faire beaucoup avec la haine, mais encore plus avec l'amour” (see this webpage).

Would you know where it comes from?

Since you say the quote isn't exact, the best I can remember is the following from Romeo and Juliet (emphasis mine):

Alas, that love, whose view is muffled still,

Should, without eyes, see pathways to his will!

Where shall we dine? O me! What fray was here?

Yet tell me not, for I have heard it all.
Here's much to do with hate, but more with love.

Why, then, O brawling love! O loving hate!

O any thing, of nothing first create!

O heavy lightness! serious vanity!

Mis-shapen chaos of well-seeming forms!

Feather of lead, bright smoke, cold fire, sick health!

Still-waking sleep, that is not what it is!

This love feel I, that feel no love in this.

Dost thou not laugh?

The above text is from Act 1 Scene 1. Romeo says this after he comes to the place where the Montagues and Capulets' servants had been fighting. The fight, he says, is more about love than it is about hatred. Even though they are representing hatred by fighting with each other, they are doing so out of the love and loyalty they have for their respective masters.

This is from the translation by Émile Montégut:

Roméo. — Hélas! pourquoi faut-il que l’amour, dont la vue est toujours couverte d’un bandeau, puisse sans yeux trouver le chemin qui mène à ses caprices? Où allons-nous dîner? — Hélas de moi! — Quelle querelle aviez-vous ici tout à l’heure? mais non, ne me la racontez pas; car j’ai tout appris. On peut faire beaucoup avec la haine, mais encore plus avec l’amour. O amour
querelleur! O haine aimante! O toute chose d’abord créée de rien! O lourde légèreté! sérieuse vanité! chaos informe de formes harmonieuses au regard! plume de plomb! fumée brillante! feu de glace! santé malade! sommeil toujours
éveillé qui est ce qu’il n’est pas! voilà l’amour que je ressens, et pourtant je n’y sens pas d’amour. Est-ce que tu ne ris pas?

_{Émile Montégut (1866). Oeuvres complètes de Shakespeare, p. 304. Paris: Hachette.}

(This is the speech from Act I Scene I that is quoted in CinCout’s answer.)

“On peut faire beaucoup avec la haine” is a poor translation of the original “Here’s much to do with hate”. In this line, Romeo means “this situation [that is, the brawl between the Montagues and Capulets] is caused to a great extent by hate” but the line also puns on “to-do” meaning “an (unwarranted) uproar, dispute, or fracas” (OED).

Romeo is speaking in the riddling mood now upon him. He means that the fray has much to do with the hate between the rival houses, yet affects him more, inasmuch as his Rosaline is of the Capulet family; that what has just passed has had reference to the animosity which divides the two factions, and has also shown him the anxious affection felt on his account by his father and Benvolio. To the latter he refers where he says, “This love that thou hast shown,” etc.

_{Charles and Mary Cowden Clarke (1868). The Plays of Shakespeare, volume III, p. 147. London: Cassell, Petter, and Galpin.}

Here are three better translations:

Il y a ici largement place pour la haine, mais plus encore pour l’amour.

_{Benjamin Laroche (1856).}

Il y a bien à faire avec la haine, mais plus encore avec l’amour.

_{François Guizot (1864).}

Ici on a beaucoup à faire avec la haine, mais plus encore avec l’amour.

_{François-Victor Hugo (1868).}