I don't know what you would use the relays for. Relays are not fast enough to provide dimming on LEDs. You should be able to control up to 6 channels of LED with the PWM pins and NPN transistors.
You need to know the power requirements of the LEDs you want to control. Usually each LED has two numbers: the Forward Voltage Drop and the operating current. If there is more that one LED in a circuit you will need to figure out how they are wired.
Excellent! Those modules run on 12V so they have current regulation built in. They draw about 1 Watt each module so that means 1/12th of an Amp each (Watts = Volts times Amps). Call it 1/10th of an amp for safety margin. Each strip of 10 will use one Amp of current.
You need a 12 Volt, 3 Amp or higher (or 12 Volt, 36 Watt or higher) power supply. Connect the minus (-) side of the 12V supply to the Arduino ground. Connect the plus (+) side to the positive side of your strips. Google "Arduino Transistor" to learn how to put an NPN transistor (or N-channel MOSFET) between the negative sides of your strips and Ground. The PWM pins on the Arduino can then control the transistors to dim the strips up and down.
