I've only won 2 Diplomatic victories and each I planned from the start and one was on Huge map and one on Mediterranean which is Large. My strategy was just Expand to as large as I possibly can to get as much trade route income as I can.
The first I won was like 3 patches ago and I played as Siam on Huge Earth on Prince, standard length, and had my capital to over 40 pop. and the first 4 or 5 cities I founded all had close to 30 pop. I was making over 500 GPT towards the end and had like 40 cities.
The most recent was Arabia on King on Mediterranean, standard length, I can't remember if this was pre or post latest major patch. But I used the same strategy of Expand as much as I can and get as many trade routes as I can.
Both games I war-mongered for the most part to help keep the A.I in check, just didn't take every single capital.
If playing a more or less peaceful game then I would try and keep 2-3 units in C.S's that are near a warmongering A.I so you can block it's units from taking the C.S if they try. Both these games I had a powerful empire so anytime an A.I declared on a C.S I just demanded they stop and they did. The only time this doesn't work is if the C.S declares Perma-War on the A.I. That's when you need to watch out because I've had several games where a C.S declared Perma-war on a powerful A.I and was gone within 20 turns. This really sucks if its an A.I more powerful than you or far away because unless you try and take the C.S back then you lose out on the bonuses.
Any other games I wanted to go for a diplo victory on ended up taking to long and I had one of the other victory conditions long before diplo was an option.