One of the pieces of advice commonly given to writers is to make sure things start happening early in your book. Writing and reading have come a long way from the days when whole chapters could pass by with just the depiction of a character's daily life. These days we need to grab the reader's attention early and hang onto it.
But the trick is balancing the need for action with the reader's own acceptance of challenge. You obviously can't put your hero into a life or death situation in chapter three and expect to incite tension in your reader, because there are still seventeen more chapters to go and it'll be pretty hard to write a story with a dead hero (presuming your story isn't about your hero coming back from the dead of course...).
That said, we still need ways to challenge the hero early on. So we need our hero to be in a position to lose something valuable. This is why a good hero needs to have a stake in what's happening. Your second chapter is too soon to put your hero's back to the wall in a showdown with his nemesis. Even if the reader believes the hero is in real danger, you're blowing too much emotion early on. Such confrontations need build-up or they'll feel wasted, throwaway scenes poorly added to pad out a slow beginning. You don't want that.
But if your hero finds out that the villain is going after a friend, or even just an innocent bystander, then you might have something. It may be to soon to kill someone off, but hurting a supporting character, that's fair game. Nothing serious. Maybe even just a scare. The villain can be chased off before any harm is done. The point is that because the supporting character can suffer without hampering the story itself, the reader can start imagining what might happen. And that is gold.
What a reader fears will happen will always be much worse than what you can come up with. Or at least, it should be worse. If you push your reader too far, showing them their worst fears for your characters, and then pushing further, you can burn them out. Ever after, when a character is in trouble, they'll look back to that horrible thing you did and reassure themselves that what's to happen next couldn't possibly be that bad. Until the moment you go even further. Before long, your work becomes an increasingly bloody trail of suffering, with you always trying to find the next fix for your reader.
Hold back a little before throwing your characters in harm's way. Don't just have bad things happen for the sake of excitement. Think very carefully about how those early threats will affect the tone and events of your book, and make sure they serve the story.