Marlon Brando did not receive an Oscar nomination for portraying Johnny Strabler in The Wild One.
Since it is Marlon Brando we are talking about here his role as Johnny Strabler is merely his least iconic out of his iconic roles. Brando plays the role of a 50's rebel, a 50's rebel being a young man who just seems to have more of a bad attitude rather than honestly fighting for or against something. He rides bikes with his gang who go harass a motorcycle race than proceeds to go harass a town for awhile. Brando plays Johnny in a fairly simple in some way one dimensional fashion, although I don't think he exactly plays his incorrectly by taking this approach. Johnny after all really doesn't have a purpose in his behavior such as when he randomly fights with the leader of another gang or frankly shown directly when asked what he was rebelling against and all Johnny replies is "Whadda you got?". Johnny is less of a rebel and more of just some young guy who just seems to not care about anything.
Brando's performance does work within the confines of his character as his mostly one note takes makes since for the character. Johnny isn't even like James Dean's Rebel without a Cause in that he secretly, or at least not according to what is shown in the film, that suggests he's pained by something in his personal life with family. Johnny on the other hand seems to be more or less content in being a motorcycle punk. Brando's performance therefore mostly depends on his charisma to portray the particular 'cool' of Johnny. Brando certainly is good in doing the understated charisma which works for Johnny. He has the power of personality and he does exude that certain element that would make everyone seem to almost bow before him whether or not they are part of his gang. Although Brando never has that smouldering magnetism that was found in his performance in A Streetcar Named Desire.
Brando's performance stays fairly steady except for a few moments where Johnny hangs out with the local waitress. Brando is fine as he suggests a little depth in Johnny as he pursues her. Brando is good in that he shows Johnny doing it in a half-hearted way as he portrays Johnny trying to hang on his 'cool' even when he is technically caring about something for once. Brando's best moment actually comes late in the film when the two finally turns on the bikers and attack Johnny. After Johnny manages to escape the mob Brando has one great moment where he completely loses the cool. Instead he rather powerfully turns Johnny into just a scared kid for a moment who is both surprised and hurt by suddenly being treated in such a way. That is a fantastic moment as Brando reveals Johnny's attitude to technically be a facade allowed by safety.
After that point Brando brings Johnny back to his normal not caring attitude although he adjusts it just enough to now suggest that he is more actively hiding emotions just underneath the surface. Brando no longer makes it a pure calm but rather a bit of cover as though perhaps Johnny could break down under the wrong circumstances. Unlike most of his other iconic roles this isn't quite a great performance. The character is perhaps a bit too limited and technically Brando's work here is not exactly filled with variety. Brando's approach does work, and his particular charm works for the character. Also the three dimensionality he does managed to infuse in Johnny comes out naturally and does make for some fairly poignant moments come from a character who simply could have been obnoxious.