(my previous experience with 'DIY' was not good, somehow on a boat it's different).--
When we bought Isabella, part of the deal we struck was that all of the running rigging would be replaced with new. This was done beautifully and we have lovely new ropes on everything (something which probably would have cost us £100 to £200 to organise ourselves) - except they did not replace the main outhaul, because to do so would have involved drilling out the rivets on the end caps of the boom, fiddling around inside, and re-riveting them back on.
The main outhaul runs up to the boom at the mast end, and dives inside over a pulley. At the other end, coming out of the aft end of the boom and over another pulley, is a length of wire, which is then attached to the clew of the sail.
Inside the boom, the outhaul rope travels aft from the mast end, passes over a pulley, then runs forward again to a permanent attachment point - as mentioned, inaccessible without dismantling the boom.
Here's a diagram:
It bugged me that this outhaul rope was getting ever greener and tatty looking - I really wanted to replace it so it matched the rest of the rigging - but I don't have a riveting gun and don't like spending money if I don't have to.
I realised that the reason the outhaul is arranged as it is in a Selden/Kemp boom as on a Tempest (and many other boats I believe) is that the boom was originally designed without having all lines on the boat lead aft to winches. To that end, the boom itself actually has cleats built into it at the mast end, for the reefing lines and outhaul. In my case, because all lines are now lead aft, I no longer have a need for the 2:1 gearing because the outhaul can be winched, rather than having to do it by hand.
So here's how to replace your outhaul:
Tie a stopper knot on the loose end of the outhaul, and pull the wire stop out of the end of the boom until you can pull the pulley completely free. I had to repeatedly pull back and forth to get it all the way out as it was getting caught on the other lines in the boom. Wasn't too hard though.
Cut the outhaul, leaving two tails sticking out of the boom, and connect the free end to your new outhaul. I used a needle to thread whipping twine through the new rope so that the pull would be central, ensuring an easy retrieve.
Pull on the old outhaul to pull the new one through and into place, then using a shackle and figure eight knot, attach the new outhaul to the wire strop.
That's it. Nothing remained but to pull it all up tight.
I left the old fixed part of the outhaul loose inside the boom. An alternative solution would have been to re-install the pulley arrangement by simply sheep shanking the new outhaul onto the existing fixed part of the old outhaul (a workable solution as the outhaul inside the boom, not affected by sunlight, was actually in excellent condition) - but in my case the pulley was seized and would require replacement, and I didn't need it anyway.