Notes from author
Depths is a very active zone. This is one of the hardest challenges I have ever done in Dark Souls.
First, you can't EVER upgrade a normal weapon above +5, and ANY special weapon at all (for example the drake sword), this is mandatory because of the weapon matchmaking system. Soul level is cap at 26 for the invasion range. Also, NEVER talk to Kaathe.
STEPS
Choose Master key as gift. First step is getting the Drake Sword, since is the only weapon you will use to beat the game, you should also talk to solaire. Kill the Gargoyles and ring the bell, do the same with Queelag with the firelink shrine shortcut. After killing Queelag kill Ceaseless Discharge with the cheese strategy. The next step is killing the Iron golem, you will need to summon Tarkus there. Next step is killing Ornstein and Smough, Solaire invocation is recommended. After that, teleport to the undead church and buy the symbol to kill Sif, proceed to kill him, I highly recommend grabbing the stone armor set. Then, kill the 4Kings, which is extremely hard with the drake sword and the stone set, this will give you access to the Darkwraith covenant and the Red Eye Orb for invasions. Put the Lord Vessel in place. The next steps are killing kirk invasions, high item discovery is recommended for the Kirk Set drop rate. First one in the Depths, the next one on the Lost izalith entrance and Lost izalith boss fight, so yes, you need to kill the centipide and the demon firesage, then you have access to the Kirk Set and can finally invade.
Final recomendations
Grab the serpent ring for high item discovery chance on kirk invasions.
Kill the Moonlight butterfly and summon the witch so you can summon her later on the 4kings.
You can also avoid ALOT of difficulties having a friend summon helping you on the difficult bosses and giving you the kirk set. I didn't, so is completely posible doing it alone, but if you can you should.
You can also cheat I guess. But I recommend trying this challenge.