Based on online pregnancy calculators, I could be 5 weeks pregnant by now. I think I had implantation bleeding on May 19 which was the day when I should have had my period. I took a home pregnancy test (HPT) today, and it turned out negative. Was it too early to take the HPT? I am planning on going to the doctor by this week for check-up, or should I wait until next week?

Most often a pregnancy test would pick up a pregnancy at this point, but not always. (you may be 5 weeks from your last period, but if you did indeed have implantation bleeding on the 19th, you would be closer to about 2 weeks of actually being pregnant). Your doctor may just do another urine test, which is no different than many you could do yourself. If they will do a blood test, ask for that.
It could be that you had an off cycle, it happens sometimes, but if you feel there is a good chance for pregnancy, see your doctor.