Dreams like this are awful and scary especially the first time. I am by no means a dream analyst, keep that in mind. I've had dreams of killing many people before and was terrified that I was a person capable of murder when I woke up. Try to think about where you are in your own life. Often times going through periods of mental/spiritual shifts in beliefs can cause you to "kill" people in dreams, people can represent old ideas that you are trying to change and be rid of. Since you only seemed to kill strangers who crossed your path it may be that you are trying to keep people from getting close to you and not allow them in in your waking life. Your dream definitely has a desperate vibe to it so I'd recommend trying to figure out whats going on in your waking life and trying to connect the two.

Remember dreams are symbolic, (you might not be a serial killer in real life )I believe its your sub-concious trying to work through issues in your waking life.

Hope this helps!