I am 18 and I'm on birth control pills since August 2014. I got my last period on March 29th and it apparently ended on April 3rd when it was supposed to, during that day I had no spotting or anything, I was all clean so I thought my period had ended. Later on, the night of April 3rd I had a little spotting, there was just a little bit of blood. The next day (April 4th) I had sex in the night and the following day (April 5th) in the morning. That day (April 5th) in the afternoon I noticed a little spotting, very little blood again, and again in the night and that would be it.

I hadn't forgotten my pills during that time. I did forgot it around a week ago (it's currently April 14th) I don't clearly remember, but I took the pill as soon as I noticed a few hours later than I usually take them. This week I started feeling a very little pain in my breasts and cramps, which I feel a week or so before my period. My following period should be on April 25th. Do I have a chance of being pregnant? This strange little "after-period" spotting is scaring me a little.

It's too hard to say, only time will tell. As long as you're fairly good taking pills, a few hours won't matter. They aren't 100% effective though, so keep that in mind.
Mid-cycle spotting can occur for many reasons, one of which is sex.

Either way, if you're a week early for your expected period, it's much too soon to know if you're pregnant. Symptoms usually aren't felt till much later and a test wouldn't be accurate for at least another week.