I think this scope has logic analyser capability. If you want to see the clk and data and protocol stuff, you should use the logic analyser to do it [As ataradov has hinted to you on I2C decoding]
But note that since it is a EEPROM, the signal will only be available depending on the user action example storing or retrieving data. The signal may not always be there or is addressed to this chip. Hope you understand.
However, if you just want to confirm signals are exchanged, use 2 probes, one on Channel 1 to the clk and Trigger on it, and one on Channel 2 to the data line. The 2 crocodile clips referenced to the ground. Or will just 1 probe, visualize the signal health will be enough for you?
So I think the proper way is to use the scope logic analyser capability to confirm its addressing to the chip, specifically.