Welcome to healthcare magic.
From your description it sound like it was infected acne
An important thing to notice in black dot/mole is to look wether there are warning sign of melanoma-skin cancer
They are alled ABCD
A for Asymmetry.One half is different than the other half.
B for Border Irregularity.The edges are notched, uneven, or blurred.
C for Color.The color is uneven. Shades of brown, tan,and black are present.
D for DiameterDiameter is greater than 6 millimeters
If you have any of these sign you should see a dermatologist