Proxy farming is one of the most complicated thing to do properly in the entire game. But, it is always fun to give it a shot. I was working on proxy farming builds for Singed (Of course), Garen and Trynd. Anyways...
So, on Singed, your first question is: Should I proxy at all. Proxy farming is strongest in one situation: You are laning against an opponent you know you struggle against, and that opponent has poor sustain. A good example is a ranged mage bully. Ryze was the classic example before they gutted him. Lissandra is a pretty good example now. You will ose to a Lissandra in lane, she has ranged harass, CC, and an escape. She also has no sustain. Other situations where might want to proxy farm include: The enemy is jungling someone weak to being pressured in their jungle AND your jungler wants to play aggressive and have presence in the enemy jungle near you (This is what SSG White did at Worlds with their Singed game).
Now, when should you not proxy? When in lane against someone you think you can beat or have kill pressure on with ganks. If you are deep in the enemy tower line, your jungler can't really gank your lane opponent. So if you think lane kills are a possibility, and you think you can beat or go even, generally don't proxy early in the game. You might still end up proxying later in the game, which is fine and a normal piece of Singed play.
Now that you have decided that you should proxy, you have a few timings to choose from. The first is at the start of the game, the second is post-6 when you hit an item/level threshold where you can kill minion waves quickly and have the movement speed to escape ganks easily and the third is after the lane phase where everyone is rotating.
If you choose to proxy at the beginning on the game, you have a scary situation. There are a ton of things to deal with. Generally, you want to start Flask, but you can actually start whatever you want. You need to know junglers very well to know if the enemy jungler will start Krugs if you are bottom side or Gromp if you are top side. If they start on the same side you are trying to proxy, they just come kill you as you carefully kite your first wave and take a good deal of damage. If they don't, you have to make sure you are gone gone gone before they get to that side of the jungle, especially if they are a high HP jungler. This means you get two waves. Ward behind you after you kill the first wave to see if the jungler is coming early. If so, you may need to suicide to tower. After you have cleared two waves, back immediately and buy. From there, you either go back to lane to take advantage of the fact that your opponent is low from tanking minions or missed CS under tower and you now have item advantage or you TP to your ward and go back to proxy farming. I like to buy a pink and drop it in brush behind me so I can see the enemy coming and suicide if needed. It is really hard to make proxy farming work at this stage of the game.
Post 6 after you have boots 2, you just clear a wave and run back to by their inhibitor and kill minions. Drop a pink and another ward or two behind you, kill waves. If their top lane and jungle try to collapse, ult and ghost and lead them on the longest escape you can possibly make: The longer they spend chasing you, the farther behind they both get. Unless they have a lot of CC or burst, you should be able to escape. Repeat for the rest of the game. The key to this working is two fold: While the enemy jungle and top (And even mid sometimes) are trying to kill you, your counterparts should be farming or making plays elsewhere on the map. The moment the jungler starts camping you top, your team should rush Dragon and shove bot towers. If your team doesn't have this coordination, proxy farming doesn't work as well. The second key is that you get ahead of your lane opponent by getting perfect farm (Because you are tanking them and can't physically miss any), while their tower is taking damage from the minions and they are trying to CS perfectly under tower. If they ever come back to try to punish you, they miss at least one wave, generally more if they try really hard. 2 waves of minions is close to the value of a kill (And worth more if your value is low), so even if they DO get you, you generally break even. If they DON'T kill you, you get a kills worth of gold ahead! But, if you don't manage to get any farm yourself, you fall behind. It is a careful balancing act.
Now, after lane phase is where things get crazy. This is when the whole team coordination thing is really important. Once you have your tank items, no single member of the enemy team will be able to touch you. This means you proxy farm HARD whenever Dragon is about to spawn (Or, proxy bottom when threatening Baron). If they send no one to stop you, proxy their friggin' NEXUS, grabbing all three waves of minions and killing them so every single lane on the map pushes stupidly hard. Crush their base while your team tries to not fight. If they do send one person to stop you, just proxy that one lane and be a general dick. If two go to stop you, your team gets the objective. This is how Singed team fights. He is, generally speaking, not very good in a 5v5. But is is suuuuper good at making map pressure for your team to take advantage of.
I can go into more detail about ward positions and stuff if you want, but proxy farming is way more about reading the enemy team and causing chaos for your own team to take advantage of than it is a specific set of items or plays.
