I have this issue where my chain keeps jumping while i pedal. It seems to have something to do with the amount of torque i put on the pedals.

I have a fairly new chain and my chain line looks good.

Any ideas on what that is?

New chain and old cassette? This will cause it to skip while pedaling but usually only in a few gear combinations...the ones you ride the most. It could also be a tight link in your chain. You may have pinched a link during the install. Check for a tight link by backpedaling the bike slowly while watching the derailer. If it's a tight link, you'll see the link as it goes around the jockey pulleys.

Umm new chain old sprocket. Its actually on a Sturmy 3sp. I put a derailer on it to take up chain slack.
Without the derailer there was never any skipping or wierd stuff so i dont think its teh sprocket.
It's the appropriate sized chain and sprocket to work with the derailer 3/16 or whatever, I figured it out when i got the parts together.

It doesnt just skip once a chain rotation though, so it couldnt be a tight link right. It skips a lot!

Another note is if im not riding my bike it doesnt skip, just pedaling it while standing beside it has no skipping.

First off you should be setting the chain tension by adjusting the chain length and shifting the 3 speed hub back and forth in the horizontal dropouts that your frame SHOULD have to use it with an internal gear hub.

But if you're using a vertical dropout frame where you can't slide the wheel back and forth then what you need is not a derrailleur but a a chain tensioner. Google for "single speed chain tensioner" to see lots of ads and pictures for options. What these do is ensure you have tension and a better chain wrap around the rear sprocket. By using a derrailleur instead you've reduced the chain wrap and that along with the used condition of the sprocket is likely what is causing your skipping.

With a proper shape sprocket the chain should not skip anyway so the skipping is a sign that the sprocket should be replaced. Why it didn't skip before is because the chain length was limited by the chain length so in a proper setup there would not have been enough length to allow it to skip. But there would be enough play to allow the chain to ride up the teeth a little. And that alone is not a good state of affairs.

So if your frame allows shorten the chain and set the play by adjusting the wheel position in the horizontal or semi horizontal dropouts. If it's a vertical dropout frame then get one of those single speed tensioners to use instead of a derrailleur. But either way it sounds like you should replace the sprocket at the same time.

And just a note that the use of the word "tension" for fixed length drivetrains is a poor word. The chain should never actually go tight so that it has any tension in it. What it means is that there should be minimal slack in the chain. I like to set my single speed chain so that at the tightest point there is still 3/8 to 1/2 inch of vertical slop at the mid point of the run. If the loosest point has even up to 1 to 1.25 inches of play to avoid the tightest spot going tight then that's fine. The tight spot can occur at any point in 5 to 8 pedal rotations when the rear and front both have their "worst" runouts coinciding so check this until you find the tightest spot. When you find it THAT is where you would clamp a fixed tensioner to allow for the 3/8 to 1/2 inch of slack. For a sprung tensioner you won't need to do that.

hmmm so, before this my chain would fall off more than I liked, in my mind this works with your analysis (which seems perfect by the way)

How do you know its not worn teeth on the derailer pulleys or soemthing like this?
I find this very hard to diagnose because it only happens while I ride, in which case I cant watch the effects well. So thanks.

Here is a picture of my sprocket and derailer

Uh, this is a 1/8 in chain (single/3sp) not 3/32, correct?

nope 3/32", its a thin sprocket on a 3sp
from a shimano 3 speed.

so does that sprocket look worn?
I've found a place that sells them.

It SEEMS OK but of course it's a rather fuzzy picture when it comes to looking for this sort of wear.

One test you can do on the rear sprocket is to pull on the chain at the very rear. If the chain and sprocket are a good match you will only be able to lift the chain a little bit away from the sprocket. If the sprocket is worn you'll be able to lift it a lot and may well see daylight between the chain and sprocket.

why the RD?
Adjust your chainline with some washers on the crank spider, or washers on the rear cog. Adjust the length of the chain (as I can see you have sort of horizontal drops). And then the chain should never jump off, like it was meant.

Arbeit macht frei? Sometimes things don't work, like deraileurs and 3 speeds...

I had a derailer and didnt have a chain tensioner. I want to use my front 3 gears.

I've seen it done a number of times here.

OK I will try that tomorrow. Excited

OK so i did this, and i can see the chain move on the sprocket. but only the half of the link that im pulling on moves away from the sproket. To form a little triangle with the 1/2 link its attached to.

The other half of that links stays in contact with the sprocket, is that worn?

https://sheldonbrown.com/chains.html
check the last part of this article to : "Chain and sprocket wear"

ok i looked at sheldons page and i have a new chain and old sproket but all my teeth seem to be perfectly centered in the middle of the links.

EDIT: I had a picture here but it sucked, I took another one.

How far should the teeth protrude? I just see little nubs. when looking at it face on.

OK i took a new picture, this is a new chain, does the sprocket look worn?
I dont think it does but would like another opinion.

Well, they used to work:

There is no reason whatsoever that a derailleur cannot be used with a 3 speed hub.

However, the skipping is likely coming from a worn cog or chainring... this is the most common cause of skipping and the wear on the cog is often not visible to the naked eye.

Another possible source of skipping under load is a mal-adjusted or damaged SA hub.

But since you said the drivetrain worked before adding the derailleur, I agree that the most likely cause is worn cog and derailleur giving the ability to skip. New cogs are cheap so that could be the first thing you try.

Next would be eliminating the derailleur and use the horizontal dropouts to tension the chain (don't use the front shifter during this test!) and see if you can make it skip without the derailleur... because that would indicate a problem with the internals of the SA hub.

IT is very unlikely it is wear in the derailleur causing this... you actually don't need teeth on the derailleur jockey wheels to use it as a chain tensioner - some chain tensioners have non-toothed pulleys and they work fine. Also, I doubt going to a single-speed specific chain tensioner will help because most of them are spring loaded just like a derailleur. Some of them are just derailleurs without lateral movement.

edit: ONe more idea: It could be possible that the derailleur is slightly misaligned and is trying to 'shift' the chain to one side or the other, causing it to ride up on top of the cog and slip. If you look at the derailleur from the rear,are the jockey wheels 100% perfectly aligned with the cog? SOmetimes the limit screws are not quite enough to get a derailleur lined up for use as a chain tensioner and you need to rig up a segment of cable to hold the derailleur in place.