To quote a German poet, Du sprichst ein großes Wort gelassen aus,
landofnowhere. Entire libraries have been written on this subject, for good old Will's influence can't be overestimated. For the first one and a half centuries, the influence wasn't felt that much, because the most that could have happened was that some travelilng theatre troops may have adopted some Elizabethan plays, but that's mainly a theory about Marlow's Doctor Faustus. Daniel Kehlmann when writing his novel Tyll decided to make Elizabeth Stuart, daughter of James VI and I, aka the Winter Queen, into an important pov character in said novel because he realised she was probably the only person in Germany (well, German speaking lands, there not being a state named Germany at the time) in the first half of the 17th century who not only knew Shakespeare's plays but must have seen Shakespeare in person, given the Lord Chamberlain's Men became the The King's Men when her father became King and must have performed for her family a lot.
Anyway, however much travelling players may or may have not transmitted between ca. 1600 and ca. 1750, the big cultural influence started (and never stopped) in the second half of the 18th century. This was when everyone, meaning a considerable part of the literary scene, started to get a wee bit tired of the cultural gorilla of the times (and ever since Louis XIV), i.e. French art, literature, and manners. Especially of the strict adherence to the three unities rule in French drama. Also, German literature started to become a thing and the Alexandrian verse used in French drama doesn't flow naturally in the German language. As opposed to English blank verse. Also, Hannover, one of the more influential and powerful German principalities, happened to rule Britain (or the other way around, but hey), which meant a lot more cross cultural pollination (and motivation to learn English if you wanted to have a career). So not only did the first few serious translations attempts of Shakespeare begin, but Shakespeare mania. The Sturm und Drang poets loved him, from young Goethe onwards. (Note: it's a bit of a deceased equine, but Sturm und Drang is not the same as Romantic. Every German teacher will get a bit of a fit if you refer to Goethe & Co. as "Romantic" in a literary sense, as the English speaking world tends to do. God knows Goethe would have had a fit. He had something of an anti Romantic bias in his later years, which is when the German Romantic literary movement was going on in, the 19th century. Sturm und Drang is 18th century.) So young Goethe, in 1771, wrote his first theoretical text about the Bard,
Zum Schäkespeares Tag, which basically goes "Fuck the three unities, Shakespeare is life, he's the best, and no one else has ever been able to present the richness of nature the way he did, if we're ever going to write decent dramas in German that needs to be our role model, and also fuck the three unities!"
This became an increasingly popular attitude among the younger writers. Mind you, the drama young Goethe wrote, Götz von Berlichingen, while clearly trying to follow up with practice upon theory, does strike one as pseudo Shakespearean and isn't one of his masterpieces. (It's JWG trying to write something like the Henry IV plays using German history, and today mostly gets renembered because the titular character tells someone to kiss his arse in it. This to many a German student in the 19th and early 20th century was ever so tiltilating.) But the amazing thing is that German literati weren't the only ones getting interested in Shakespeare. So did people like
Ulrich Bräker, self educated farmer's boy who responded to being gang-pressed into Prussian military service by deserting in the very first battle of the 7 Years War at the first chance he got (good for him), and later didn't solely write his memoirs but also a commentary on Shakespeare's works where he compares Henry V, the quondam Prince Hal, to Frederick the Great and doesn't mean it as a compliment. (To either man.) (These commentaries are really interesting to read because they are basically an 18th century guy blogging about Shakespeare's plays without knowing anything whatsoever of what other people have said about them.)
Come the 19th century, Shakespeare Mania proved it had come to stay, not least because we got our first big classic translation (the earlier attempts, including the ones Ulrich Bräker must have read, are respectable but not great), a masterpiece of literature in its own right even though it does bowlderize the sex jokes a bit, to wit, the Schlegel/Tieck translation. (So named after two of the primary Romantic - yes, this time really Romantic - writers involved in same.) BTW, this did not mean people stopped translating Shakespeare into German. They still haven't stopped. And there are a few other translations regarded as masterpieces by now, like, say, some of Stefan George's takes on the Sonnets. Also the advantage for translators is of course they can translate Shakespeare into the German of their present, whether that's early 19th century German like the Schlegel-Tieck gang or current day German. And they do.
Anyway, along with all the translations, there were the increasing number of performances. Soon every major ensemble had at least one Shakespeare play per season in their repertoire. Also: what was performed were the actual plays. (Theatrical history minded Germans are still a bit proud of the fact Shakespeare got performed in the original - translated - form while in Britain they were still staging the rewrites with happy endings for Lear or additional scenes for Romeo and Juliet where Juliet wakes up and has dialogue with Romeo while he's dying etc.) And all this performing and translating also meant Shakespearean subjects for painters and composers, most famously of course Mendelssohn.
Sadly, the 19th century was of course also the time when nationalism became en vogue and rose to an ever increasing fever pitch. This led to some surrreal and bizarre attempts at Shakespeare theory trying to prove he really must have been German. (I mean, better German than the bloody Earl of Oxford, right?) And that Hamlet, the character, is really meant to symbolize the German soul. (Don't ask.) But even leaving these distortions aside, the saner way of influencing merrily continued. If you want an example of non-nationalistic 19th century literary Shakespeare criticism, try Heinrich Heine.) And if you want an example of a later famous composer tackling the Bard in his struggling days when still learning the ropes, check out young Richard Wagner making an opera out of Measure for Measure, with some interesting rewrites. This play is actually set in Vienna, but Wagner transports it to Sicily. And it ends not with the Duke unmasking himself etc., but with a popular uprising against Angelo (that's Wagner the 1848 revolutionary for you); also, Isabella ends up not with the Duke (and not as a Nun), but with Lucio.
Come the late 19th and early 20th century, an argument could be made that the globally best known and most popular Shakespeare production was Max Reinhardt's Midsummer Night's Dream, not least because he did go on a world tour with it. (Decades later, as an old man and exile in Hollywood, he also did a movie version. But Reinhardt was a theatre director - in his day and for at least two decade, THE theatre director - , and cinema wasn't his medium.) And the fact that by then, of course there were plenty of German plays to perform never stopped the contiued success and demand for Shakespeare performances, or the fact his works keep getting reinterpreted and projected into by every era and by a lot of other artists. During the Third Reich, you had, one the hand, several absolutely vile productions of The Merchant of Venice, obvious reason is obvious. But you also had a production of Richard III, directed by Jürgen Fehling and produced by one of the most prominent theatres, the Theater am Gendarmenmarkt, which according to many a description went for a "demagogue murders his way to the top, becomes evil dictator, is eventually toppled" direction that between black leather for Richard's henchmen and Richard evoking the most prominent current hunchbacked, lame footed guy in politics, i.e. Goebbels. (Mind you, those descriptions were written retrospectively. You can bet no current review mentioned something like that, though then again, they wouldn't have. And there are stage photos.) Richard was played by Werner Krauß. Who also played Shylock in those antisemitic Merchant productions, and notoriously every Jewish character except the title role in Jud Süß. (Krauß had had a long and distinguished career before the Nazis. Being a great actor does not, alas, mean being a good or at least semi-decent person.) And there you have the ambiguity of German Shakespeare interpretations in the worst time of German history.
(Shakespeare was never cancelled, btw, though some current day English writers were in the last WWII years. Some other stage hits originating in Britain like Oscar Wilde's plays got excused by declaring that hey, Wilde was Irish, but to my knowledge no one even tried demanding a justification as to why Shakespeare was still put on the German stages. He just was.)
These days, Shakespeare is subjected to the same Regietheater extravaganzas as every other classic and continues to be newly translated. I think the Histories have gone out of fashion - except for Julius Caesar, that's still performed a lot - , but otherwise the same plays are popular or less popular as they are on English speaking stages, and poets keep tackling the sonnets. And we have Oxfordians, God help us. (About fifteen years ago, a director produced Macbeth in my hometown titled "Macbeth, by Edward de Vere, "Shakespeare".) Not to mention that Roland Emmerich is from Suabia. I don't think anyone, not even the loathsome bunch of current neo nationalists, tries to claim the Bard was really German, though. (Progress?) Though you do occasionally, just occasionally, come across a writer admitting they like the Schlegel/Tieck translation better than the original... Der Rest ist Schweigen.
The Other Days