What does the alignment rule mean if you have no deity?
It means absolutely nothing to you, it will only affect which spells you can cast, as you cannot cast spells of an opposed aligment.
I am currently Chaotic Good; would that mean that I cannot become
Neutral Good because I don't have a god to give me an axis?
No, you could become Neutral Good. The fact you don't have a deity doesn't affect what aligment you can be, or change to. The "axis" isn't exclusive to a deity, it is about the lawful/chaotic and good/evil axis, and it's the same to everybody.
You can be a cleric of no deity, but you must follow an ideal (choose two domains), the aligment in this case will change what type of energy you can channel (positive or negative).
If you are a good cleric, you can only channel positive energy.
If you are an evil cleric, you can only channel negative energy.
If you are neutral, though, you can choose (when you take your first cleric level) if you will channel positive or negative energy.
Your lawful/chaotic axis won't change your powers, only what kind of spells can affect you or not (ie: protection from chaos). And what kind of spells you can cast, a lawful cleric can't cast protection from law, just as an evil cleric can't cast protection from evil.
Keep in mind though, that in Golarion (Pathfinder's default campaign setting), you cannot be a cleric without a deity. The closest thing to that is an oracle.