Why would we need to see the records of a dead young man? He's not on trial here.
The short answer is that we don't, but some explanation will be helpful. The rules of evidence do permit a defendant claiming self-defense to introduce evidence of the violent character of the victim. So to the extent that any of this stuff from Martin's past indicates a violent disposition (I don't think it does at all, for the record), it could be considered relevant. Certainly it would be relevant in public discussion if the evidence actually indicated that Martin was a violent person.
However, even if Martin had a long history of starting fights, only opinion or reputation testimony would be admissible at trial for the purpose of showing that the victim had a violent disposition. No evidence of specific alleged acts is admissible for that purpose. Also, no extrinsic evidence would be allowed for that purpose anyway, so the records themselves could not be introduced under any circumstances. What the defense could do is call a witness to testify that Martin had a reputation in the community for being a violent person, or to testify that based on knowing Martin, his opinion is that Martin is a violent person.
But if the defendant introduces evidence that the victim had a violent disposition, that opens the door for the prosecution to introduce evidence both of the victim's peaceful character (to rebut the defense witness's testimony), and also testimony about the defendant's violent character. But again, only opinion or reputation testimony would be allowed; no evidence of specific acts, or extrinsic evidence (like documents or official records) would be admissible (at least under the rule in question) to suggest that Zimmerman has a violent disposition. (Prior convictions and other specific bad acts of the defendant are sometimes admissible under another rule, but only under limited circumstances and often only for limited purposes.) The danger of unfair prejudice is the reason for all of these limitations. The rules about this might seem a bit strange, and they do exclude evidence that is logically relevant. But the rules reflect the determination that some evidence, even though it meets the (very permissive) legal definition of relevance, is likely to be given disproportionate weight by a jury.
So this stuff from Martin's record is legally irrelevant, except perhaps to the extent that a past incident of maybe breaking into a locker at school makes it more believable that Martin was doing something particularly suspicious-looking on that fateful night in February. I don't see how that would work, since even given a past burglary (which I am assuming here for the sake of argument), the only way that would bear on Martin acting suspiciously that night would be if he were looking to commit a burglary, or some other crime. People who have committed a burglary in the past don't look any different from other people if they are just walking home. I see no reason to believe that Martin was looking to commit a burglary on his way home from the store (iced tea and skittles are not only ineffective burglary tools, they would be a hindrance).
If the suspicion was based on actions that were in fact innocuous (even if they might legitimately appear suspicious to someone on the lookout for burglars), then Martin's past trouble would be entirely irrelevant to whether Martin did anything that might warrant suspicion, or whether Zimmerman's suspicion was reasonable.
Zimmerman's testimony of exactly what he saw Martin do will be very informative regarding how reasonable (or not) his suspicion was. Just because we don't have that testimony yet is no reason to grasp at things like Martin getting in trouble at school, as if that can support the conclusion that Martin was likely acting in a suspicious manner. Any facts that reasonably bear on the likelihood of Martin reacting violently to being followed would be fair to bring up, but I haven't seen any such facts.
Remember, though, that as a matter of law the self-defense claim does not depend on whether Zimmerman's suspicion was reasonable. Practically speaking, though, it may affect the jury's opinion of Zimmerman's character, and that might affect their opinion of his credibility. But I think that would only have a major influence if Zimmerman
totally bombs on the witness stand and comes across as an overt racist or something.