The most obvious way he might get caught is for someone to find the Death Note. Unfortunately, that's also the hardest to avoid. He could hide it at his home, but no matter how well he conceals it, it might still be found. If the police search his home while he's away, they might find it no matter how well he hides it. For that matter, his father and his sister both have a key; it's not likely they'd look through his private possessions, but not completely impossible.

On the other hand, if he keeps it on his person, then a search of his person will almost certainly find it. But he might have warning, at least, and the warning could be enough for him to hide it. 

Really, the thing to do is to try to come up with a way to make it look less incredibly suspicious. A notebook full of the exact names and causes of death of every past criminal he's punished and several future ones is as good as a confession; a notebook that says "Death Note" on the cover, with the rules on the inside, is -- still a risk, if they figure out that's how the criminals have been dying, but a risk that he has a chance of explaining away.

But he can't "make it unusable by tearing it or burning it up," and he's not sure how much he wants to gamble on that being the exact wording that matters. Is he safe if he makes it unusable by submerging it in water? Is he safe if he burns a single page? 

Eventually he settles on carefully cutting out a single page and dipping it in ink so it's unreadable. For the sake of the experiment, it has a single name on it, and he waits until they're already dead. His heart is pounding -- this really ought to be safe, but it still feels like a gamble -- but he finds himself just as alive as he was before.

(He feels vaguely disappointed, for some reason, at the result of the experiment. How strange. It's not as if he wants the rules to be written unfairly.)

On his person, then. He'll cut out every page once everyone on it has died and dip it in ink until it's unrecognizable, and keep a bottle of ink with him. He doesn't know what'll happen if he has to destroy a page where not everyone is dead, and there's no ethical way to test it that doesn't carry some risk of getting caught. Hopefully it won't come up.

The second most obvious way he might get caught is for him to make a stupid mistake that gives himself away. He will ... simply not do that. It's honestly strange how so many criminals pull off crimes near-perfectly, then happen to get caught on video camera in the act. 

Still, he takes some reasonable precautions. He uses heart attacks consistently for deaths that he wants associated with his project, and doesn't get too creative with the details; if he needs to kill anyone else, if there's a repeat of the Miyanagi incident, he'll use some other method, and no one will be any the wiser.

...Okay, maybe he'll make one stupid mistake. One is safe, and -- this is important.

灰根Haine 高太郎Kōtarō — Commits suicide, alone, at 3:18 PM on April 14th. Leaves a suicide note, in his own handwriting, mentioning nothing related to the DL-6 incident.

(In his dreams, his hand brushes against the trigger of a gun, just for a moment. When he wakes up after barely two hours, it takes him a moment to remember where he is.)

The third most obvious way he might get caught is for someone to look at his search history. Deleting it is useless, maybe worse than useless if they think he has something to hide.

They won't check, without a reason -- sensible though it might seem to monitor everyone's internet history, there is far too much activity on the internet for this to be feasible -- but if they do check, it'll look suspicious at best. He's a prosecutor, yes, but even a prosecutor has no reason to coincidentally look up the names and photographs of everyone being punished by Kira. (That's what they're calling him, now. He doesn't really like it; he'd prefer justice, even revenge, but he can hardly say that out loud.)

He compensates by developing a "hobby" of consuming true crime fiction. He carefully looks up names related to the cases, and other criminals, living and dead, similar to the ones described. It's far from a perfect plan, but better than no plan at all.

...It's unexpectedly painful to read about a criminal who got away and not be able to punish them, but it would kind of defeat the point if he did. (He hesitates for half an hour over causing another suicide, but he can't justify the risk.)

He's careful not to let on anything at work. People notice that he's tired, maybe. It's a failing on his part to give away even that much, but he doesn't think it's a suspicious one. It's not like it takes him more than a couple minutes to handle the killings, and he's writing them well in advance lest something take him out of commission for a few days. It's not connected, so it shouldn't be evidence of guilt.

He tells them he was up late reading. It even has the virtue of being true.