Hey,
It took me while to get used to deity too. It's important to remember that you are playing catch up (but don't get too stressed about it), so I find it easier to play it passively at the start. Try to get your scouts out, find as many AI as possible (for the science bonus and trading partners), and identify possible threats (aggressive AIs). If you have anyone close, make sure to put 3-4 archers and just defend yourself. Otherwise, try to keep them busy by bribing them to fight other AIs (I find that keeping AIs fighting each other really eases your way to victory).
I like to go for tradition and trying to grow the cities 3-4. While you always feel the need to keep building things, it's important to identify which ones are useful and to keep an OK army. Rationalism will help you with the science, and sooner or later you'll find yourself to be even or ahead of the AIs. Remember to use the research agreements. I like to build porcelain tower whenever possible (for SV).
Careful with the ideology, since it's likely you'll get a lot of pressure if you don't have a big culture pool. Often I like to invade my neighbour to acquire the wonders that he has built (which give me more culture and happiness, as well as increase my overall strength).
if you have kept your army up to the level, you should not have too big of a problem in the late game.
However, if you really are struggling I would recommend you to watch any Let's Play Deity videos that will give you a grasp on what to expect and how to react.