The best place to start with answering this question is the text itself and the language features that Miller gives to Biff elsewhere in the play. What your teacher probably wants you to do is to analyse how Biff speaks in the play, and to use similar features in the diary entry that you are being asked to complete for your assignment. One thing about Biff that Happy notices is that he does have the habit of speaking in quite lyrical langauge. Happy calls him a "poet" when he says the following quote about how much he loves working on a farm:
This farm I work on, it's spring there now, see? And they've got about fifteen new colts. There's nothing more inspiring or--beautiful than the sight of a mare and a new colt. And it's cool there now, see? Texas is cool now, and it's spring.
If we spend a moment analysing this speech, we can see that some of the features are the kind of slang and colloquialisms of the time. Biff has the habit of using phrases such as "see" to end his sentences, turning them into questions. Also, Biff has a love of nature and the great outdoors that it would be good to reflect in his diary. What he finds oppressive is the rat race that he feels his father's ambition is driving him towards. If we look elsewhere in the play, Biff uses phrases such as "Kid, "Boy," and "Gee." These are the kind of slang features that your teacher will be looking out for, in addition to a somewhat lyrical strain. Good luck!