Best Classic Shakespeare adaptation?

Unquestionably, Zefirreli's ROMEO AND JULIET.
Beautiful film, like looking at Renaissance art.
And the two young leads, though inexperienced, were nonetheless just right.

The 1968 Romeo and Juliet
The 1948 and 1996 Hamlet
1944 Henry V

If you include those movies derived from Shakespeare play

West Side Story - from Romeo and Juliet
Ran - from King Lear
Throne of Blood - MacBeth

I honestly haven't seen that many but I really enjoy Julius Caesar (1953).

Others that I've seen:

Romeo and Juliet 1968
Merchant of Venice 2004
Coriolanus 2011

The ‘55 version of Richard III and Branagh’s 1996 adaptation of Hamlet.

Which Macbeth? Welles?

Here is a list I had drawn up sometime before but don't remember where I have shared it. Forgive me if I repeat myself.

“I love Shakespeare. He wrote some of the rawest stories, man. I mean, look at Romeo and Juliet. That's some serious ghetto shit." - Tupac

BEST: Julius Caesar / Joseph L. Mankiewicz (1953). In the lead are two British actors, James Mason and John Gielgud as Brutus and Cassius. In what some considered stunt casting, Marlon Brando in only his fourth film got the call as Marc Antony. Brando triumphs, proving once and for all that he could act in more than modern drama and that The Method was viable even in Shakespeare - and this has proved to be so to this day (see also Al Pacino, below). DVD

Titus / Julie Taymor (1999). Taymor makes a case for one of Shakespeare’s earliest and most reviled plays. Taymor ups the visual interest to 11 while retaining much of the dialog. The cast is near perfect. Anthony Hopkins is the title character. Jessica Lange who will chill you, thrill you and fulfill you as the evil Queen Tamora. DVD

Romeo And Juliet / Franco Zeffirelli (1968). This film had a profound influence on a generation and beyond. Not only did it turn on its generation to Shakespeare, it replaced “Julius Caesar” as the go-to play to teach in high school. DVD

Romeo + Juliet / Baz Luhrman (1996). R&J for the next generation. This film startled and shook up some Shakespeare purists, but, for me, it is a real thriller and truly tragic (made even more so – and more cruel - by a slight tinkering with the final death scenes.) DVD

The Taming Of The Shrew / Franco Zeffirelli (1967). La Liz and Dickie (for the younger set, that's Elizabeth Taylor and her husband Richard Burton) in Shakespeare's anti-feminist romp (or is it subtly subversive of the patriarchal society of the Early Modern Period?).

A Midsummer Night’s Dream / Peter Hall (1968). Excellent film of an RSC production with a top-flight cast. Helen Mirren, David Warner, Michael Jayston, and Diana Rigg as the four lovers from Athens, Ian Holm as Puck, Paul Rogers as Bottom, and best of all, Ian Richardson and Judi Dench as the sexiest King and Queen of the Fairies that you will ever find. Can’t miss. DVD

A Midsummer Night’s Dream / Julie Taymor (2014). This is a film of a production at the Theater for a New Audience in Brooklyn in 2013. It was shot with multiple cameras over four nights and then edited like a regular movie. Highly recommended, especially if unfamiliar with the play. A good time will be had by all. DVD

The Merchant Of Venice / Michael Radford. (2004). Al Pacino gets his chance at a full-length Shakespearean movie role and he nails it so completely that he may have ruined it for every other actor. “Merchant” is the Bard’s most controversial and disturbing play because of the portrayal of the Jewish money lender, Shylock (Pacino), in Venice. DVD

Much Ado About Nothing / Kenneth Branagh (1993)
Much Ado About Nothing / Joss Whedon (2012)

The world’s first rom-com and one of my favorite plays is given two top-notch film productions, just about 20 years apart. DVD

Twelfth Night, or What You Will / Paul Kafno (1988). Although Kafno is credited as director of the television film, what is being performed is a stage production directed by Kenneth Branagh. Branagh makes more of a serious dramatic play from the materials than is usual. I have seen a couple of university stagings that were hilarious, but the Branagh’s actors find a more serious, perhaps melancholy, path. Hard to find DVD.

The Hollow Crown.

Richard II
Henry IV parts 1 and 2
Henry V
Henry VI parts 1, 2, and 3
Richard III

The BBC undertook a series of Shakespeare’s War Of The Roses historical chronicles and has turned in a major triumph on every front. “Richard II” stars Ben Whishaw as Richard, Rory Kinnear (The Imitation Game) as Richard’s usurper, David Suchet as York, and Patrick Stewart as John of Gaunt. Tom Hiddleston carries the next three plays as, first, as Prince Hal who cavorts at the Mermaid Tavern with Sir John Falstaff (Simon Russell Beale) but who converts into the heroic King Henry V. Capping the series is Benedict Cumberbatch’s definitive Richard. He is so great that I get goose bumps just thinking about it. DVD

Hamlet / John Gielgud (1964)
Hamlet / Gregory Doren (2009)

In 1963 Richard Burton attacked the title role in a Broadway production directed by John Gielgud. This movie is a film of one of the actual nights with the sounds of the New York audience very much in evidence. Burton’s voice and portrayal are very muscular and masculine. The Gregory Doren film, starring David Tennant (the tenth Doctor on “Doctor Who”) and Patrick Stewart as King Claudius, is a film of a stage production that played to critical acclaim and sold-out houses at the Globe in Stratford and London’s West End in the Fall. Nothing is done by rote. Every character, his/her motivation, thoughts and feelings have been carefully re-thought in light of modern psychology and acting styles. Tennant gives us a Hamlet for this time in history. Patrick Stewart is the best King Claudius I have ever seen. Period. Very, very highly recommended. Not sure if the Gielgud is still available on disc. DVD for Doren.

King Lear / Edwin Sherin (1974). This was filmed live in front of an audience as part of Joseph Papp’s New York Shakespeare Festival at an outdoor theater in Central Park. James Earl Jones plays the title King and, as far as I am concerned, he is definitive until proven otherwise. Raul Julia makes a fine sarcastic villain in Edmund. Rosalind Cash plays the eldest evil daughter, Goneril, while Lee Chamberlin (most well known for her seasons paired with Morgan Freeman on “The Electric Company”) essays Cordelia. Paul Sorvino is Gloucester; Rene Auberjonois, Edgar. As in all productions I admire, there is truthfulness in acting (in a modern sense) and the text is respected. Wonderful. DVD

A Performance Of Macbeth / Philip Casson (1979). Ian McKellen steps up as the noble Thane who is tempted into murder. His performance stands with one foot in the older declamatory style but with another in modern style, especially in those moments of heightened tension. Roger Rees is a standout as Malcolm. Ian McDiarmid (Emperor Palpatine!) doubles as Ross, and scores as the Porter in the famous comedy interruption that comes between the King’s murder and its discovery. Judi Dench, though, as Lady Macbeth is the true revelation of this production. It is magnificence, indeed. She digs into the character with all her ambiguities, forcefulness, doubts and guilt. DVD and YouTube.

The Tempest / Julie Taymor (2010). In switching the protagonist’s gender from Prospero to Prospera and getting Helen Mirren to play the part, Taymor came up with a great idea which works. One reason is that turning Prospera/Miranda into a mother/daughter relationship makes a lot of sense and fits the lines they are given to speak. Mirren is terrific as are fellow Brits Tom Conti, Alan Cumming, and Ben Whishaw. American actors David Strathairn and Chris Cooper comport themselves like experienced Shakespeareans, which they may very well be. DVD

Much Ado About Nothing 1993
The Taming of a Shrew 1967, and I have to agree about 10 Things I Hate About You 1999 with OldAussie
Romeo and Juliett 1968
Julius Caesar 1953
Hamlet 1996 and the Soviet Gamlet 1964
Ran 1985

In no specific order.

I highly recommend almost any of the Shakespeare plays done by the Stratford Festival in Toronto. 
store.stratfordfestival.ca/
I have tried many of their DVDs and have yet to be disappointed.
These are all live performances with live audience reactions, as well as being very well done.

It is amazing to see King John as staged and played by this group.
Other Shakespeare plays they have done include:
King Lear - 2015
Romeo and Juliet & Timon of Athens (double pack)
Adventures of Pericles - 2016
Taming of the Shrew - 2016
I've also seen: 
The Tempest -2010 (with Canada's own Christopher Plummer) 

Then they have
Love's Labour's Lost - 2017
Hamlet - 2016
Macbeth - 2017

The list just goes on and on......

This looks like a great deal...
store.stratfordfestival.ca/collections/video-stratford-hd-series/products/stratford-hd-8-pack-series

Thanks for the kind words. 

I had only seen Olivier's Richard once before many years ago and hadn't cared for it at all. But in 2016, the 400th anniversary of Shakespeare's death, I tried to see at least one production of the 36 plays in the First Folio plus Pericles and The Two Noble Kinsman. In addition, to see or revisit as many plays and productions as I could get my hands on. Thus, I determined to watch Sir Larry's Richard III again. This time, I really got into his performance. His Richard is not just evil, but enjoying it (until, of course, the visit from the ghosts). I almost laughed out loud at a couple of places. Ian McKellan's Richard III is also worth a visit even though the text of the play has been cut to the bone. The big laugh line there (the play is set in the 1940s) comes at the final battle. King Richard's jeep get stuck in mud resulting in him yelling, "A horse. A horse. My kingdom for a horse!"