Foxxiest of the Hounds achievement requires to finish the game without sounding an alarm.
Alarms are sounded by enemies activating an alarm panel or security systems going "Hostile". Security systems going "Suspicious" don't count as an alarm so getting spotted for a split second won't affect your achievement.
Also, getting spotted by an enemy or more but without giving them a chance to sound the alarm also won't affect your achievement.
The way it works is that during certain main story missions, so side quests don't count, you can get an exerience bonus with the title "Smooth Operator".
The problem with Smooth Operator is that for example leaving a dead body where other guards may find it, sounding the alarm, will affect your achievement even if the alarm is given AFTER getting Smooth Operator. Just make sure to hide yuour victims.
By reading the missions or quests tab you will notice they are marked with letters and numbers like M1, M2, M3 and so on. These are main story missions.
List of missions to get Smooth Operator on: Detroit Police Station
M5 - Investigating the Suicide Terrorist
1. Retrieve the dead terrorist's Neural Hub Highland Park
M10 - Following the clues in Highland Park
2. Infiltrate the Mercenary Compound
3. Locate the Mercenary Leader Tai Yong Medical
M1 - Searching for Proof
4. Enter TYM's tower by passing through the Pangu
5. Find and view Van Bruggen's recording
M2 - Entering the Dragon's Lair
6. Confront Zhao in the Penthouse
7. Escape and get to the hangar
8. Open the hangar bay doors Picus
M1 - Confronting Eliza Cassan
9. Escape the ambush by reaching a lower floor
10. Get to the funicular
11. Summon the funicular
12. Get inside room 802-11Detroit - during your second visit
M2 - Finding Isaias Sandoval
13. Obtain Sandoval's location from Taggart.
14. Find and confront SandovalHengsha - during your second visit
M3 - You scratch my back, I'll scratch yours ( ONLY PART OF DLC
15. Rescue Tong's son
M4 - Stowing Away
16. Retrieve Tong's Package from the shed
17. Place the C4 on Administrator Wang's deskOmega Ranch
M1 - Rescuing Megan and her team
18. Search for Signs of Megan's Team
19. Disable the Signal Jammer
20. Upload a virus to the security computer and enter the restricted areaPanchaea
M1 - Shutting Down Darrow's Signal
21. Get inside the tower and disengage the lockdown
22. Reach the Broadcast Center
As you can see there are 22 main story mission objectives that require Smooth Operator. However, some of them excluded by finishing the objective by talking your way and so gaining a different experience bonus, "Silver Tongue". An example of this is mission M5 - 1 during your first visit to Detroit, at the start of the game, Investigating the suicide bomber. By getting Silver Tongue you will not have 21 objectives (22 with the DLC) but 20 (21 with DLC)/ This will not
affect you in any way towards the achievement.