If Skyrim decided to be hilarious, type this into the console: setstage ak69katanapersonalquest NUMBER

First stage - 1 game hour - "So, what brings you to Skyrim?"
Second stage - 3 game hours - "I'm so curious about why you're here."
Third stage - 3 game hours - "Katana, I..."
Fourth stage - 6 game hours - Upon talking to Katana, she'll tell you that you both need to head to Whiterun, which means you shouldn't be in that area for this stage. 

Talk to Katana right after recruiting her to start the quest timers.

Quest Stages

10
Description: Katana has a plan, but it looks like I need to wait and talk to her later.
Objective: Talk to Katana another time

20
Description: At the Drunken Huntsman, I'll need to sit at one of the tables and someone's supposed to come find me.
Objective: Head to the Drunken Huntsman

30
Description: Alright, I'll bite. I sat. I'm not really sure what's happening, but I want to know what Katana has planned.
Objective: Wait to be approached (and also optionally talk to Katana)

40
Description: To collect the bounty, I need to go to Silent Moons Camp near Whiterun, and defeat Lucky Irnsvar.
Objective: Defeat Lucky Irnsvar

45
Megara dismounts in Markarth.

50
Description: I've defeated Lucky Irnsvar. Now, I need to go sit at a table at the Silver-Blood Inn in Markarth, and tell Megara.
Objective: Head to Silver-Blood Inn

60
Description: I suppose sitting at taverns is code for something for these people.
Objective: Wait for Megara

70
Description: Megara gave me a letter that was found near the bandit I defeated. I don't know why she gave the letter to me, but I should read it.
Objective: Read Lucky Irnsvar's letter (and also optionally talk to Katana)

80
Description: A bandit called Galtun Bold-Thief is waiting to challenge me. I must go to Kolskeggr Mine near Markarth to defeat them.
Objective: Defeat Galtun Bold-Thief

85
Megara dismounts in Riften.

90
Description: Galtun Bold-Thief has been defeated. I need to meet Megara at the Bee and Barb in Riften and let her know.
Objective: Head to the Bee and Barb

100
Description: I don't understand why I have to keep sitting.
Objective: Wait for Megara

110
Description: Megara gave me another note. I should read it.
Objective: Read Galtun Bold-Thief's Note (and also optionally talk to Katana)

120
Description: I need to go to Broken Helm Hollow near Riften to defeat Runir Wulfhart.This might give us information to find the thief.
Objective: Defeat Runir Wulfhart

125
Megara dismounts in Winterhold.

130
Description: I've defeated Runir Wulfhart and need to go collect my reward from Megara. She also might have more information for me.
Objective: Head to the Frozen Hearth

140
Winterhold scene.

150
Megara and Katana say bye for now in Winterhold.

160
Elli approaches.

170
Description: What was that? I really should talk to Katana.
Objective: Talk to Katana

180
Description: We must go to a house near Fort Fellhammer when we're ready to face River.
Objective: Head to the location

You will either get a force-greet from Katana or you can talk to her, near the hideout's door.

190
They just entered the hideout.
Objective: Confront River

200
Combat begins.

210
River has entered bleedout.

220
River tells Katana to stop.

230
Scene after paralysis.

240
River portal.

250
No more paralysis.
Objective: Talk to Katana

260
Finale, and Katana talks.

270
Description: River got away, but it looks like Katana has made peace with this.

300
Quest stops.