There's already a fantastic walkthrough posted here on TA and I would like to link to that first and foremost: Sherlock Holmes: Crimes & Punishments Walkthrough
Here is my own take on how to complete the case - this is the 4th case that develops throughout the game taking place after Blood Bath is completed. Once the cutscene ends, look down and talk to Toby for progress in the Dog-fancier achievement. Next, examine the letter on the nearby desk and identify the address, seal and flip over for the coat of arms. This kicks off another cutscene and afterwards you can select to travel to Abbey Grange in the journal.
Once you arrive and the cutscene ends, open the doors straight ahead and enter the next room which will start some dialogue with Lady Brackenstall. Be sure to identify the fresh bruises, pale cheeks, gold brooch from Australia, elegant dress, wedding ring and old bruises to complete the character portrait. Continue talking and select "Is something missing" then "old bruises". Next, examine the newspaper on the table opposite the couch. Go to the other table to the left and examine the front before removing the back panel, examining the inscription and taking the photo. Now use Talent to investigate the photo up and to the right to see scratch marks near its corner. Examine the picture to reveal a safe but we need some more info first.
Turn around and speak with the maid, Theresa. Now identify the wrinkles, working hands, caring (handkerchief) and coffee stain to complete her portrait. Then speak to Lady Brackenstall again to get information about the safe. Return to the safe and turn it to R-15, L-5, R-17 to open it up and reveal a medical report. Grab this and examine the cash and coins before exiting the room and entering the doors opposite to the murder scene. Use talent just above the fireplace and to the left to cut some bellrope and examine it. Next examine the different glasses on the table opposite the fireplace and then the rope on the chair to the left of this table. Now continue this direction and examine the items on the wine rack counter then finally check out the body. Start by examining the feet, the stick (next to right hand), the fireplace grate, the head wound and the poker (next to head). In the deduction screen link Two Glasses + Glass with Beeswing, Dead Body + Fireplace Grate, Dead Body + Bent Poker. Make your way back to Baker Street to mull things over.
Once back, go over to the Analysis Table and Examine Rope to figure out how it was cut. Try scissors first (nope) then the knife (yup) which unlocks a new item in the deduction screen so go there and link Sharp Knife + Knotted Rope. That's all for here so talk to Toby and go back to Abbey Grange with your dog riding along. You will then control Toby so follow the green smell trail to the garden shed, the well then the stone wall through the grass which lets Sherlock piece together the journey. Enter the shed and examine the tool box, the bags and the suitcase then take the hook on the left. Now go to the well and open it so you can examine the glittering object at the bottom. Move the bucket, attach the hook and use the winch to lower the hook and bring back up a bag. Examine the back of the fork on top for the coat of arms then examine the wall which the intruders fled over. On the deduction screen link Silverware Found + Criminals Identified then make your way back inside the manor.
Head inside and talk to Lestrade, Lady Brackenwell and Theresa for the final clues. Open up deductions again and link No Personal Life + From Australia, Violent Behavior + Inspector's Tale then the following deductions; Two People, Imitated Robbery, Randalls Blamed, Poker Blow, Domestic Violence, Sailor Background and Acquainted with Sailor to get Murderous Visitor and then the task to "Look for Sailor". Go on back to Baker Street and check the newspapers next to the fireplace for Rock of Gibraltar under 1893. This will lead to a cutscene and the new task to "Search for Sailor Suspects". Go to the window and Call Wiggins who will fetch a register after another cutscene which you can then find on the desk to your left. Select the 1st page of the documents on the left to strike Henry Southward and Ernest Woods, 2nd page to strike Charles Chill and William Partridge, 3rd page to match up Jack Crocker and lastly the 4th page to strike Herbert Whittington and Thomas Walker.
Talk to Wiggins again who will bring Jack Crocker to Baker Street and initiate dialogue. Be sure to identify honest (eyes), strong build (shoulders), sea knife (belt), newspaper ink (right thumb) and clean boots. After more discussion, you'll be asked to present evidence so select Sailor's Knots but he will still end up leaving. Afterwards, on the deduction screen combine Crocker Statement to Captain Crocker then the newly created Crocker's Alibi. This will allow you to decide all outcomes of the case if you are chasing the A Complex Mind achievement by choosing the following:
Three People, Robbery is the Motive, Randall Gang, Poker Blow, Sailor Background, Domestic Violence, No Acquaintance, Crocker's Alibi --> Randalls
Two People, Imitated Robbery, Randalls Blamed, Deadly Accident, Sailor Background, Domestic Violence, No Acquaintance, Crocker's Alibi --> Accident
Two People, Imitated Robbery, Randalls Blamed, Poker Blow, Murderous Visitor, Domestic Violence, Sailor's Background, Acquainted With a Sailor, Look For Sailor, Crocker's Involvement --> Crocker
When you go to accuse him, be sure to save him by selecting his hand and completing the on screen QTEs to earn the Hold Your Herrings, Captain achievement. After the cutscene, you'll finally earn the Vox Populi, Vox Dei achievement.