Darius is generally an excellent solo top champion due to his durability and harassment ability, and he can easily gain lane control against most other melee tank/bruiser opponents.
Decimate is a powerful harassment ability. Try to strike enemy champions from maximum range for the greatest effect.
Liberally apply Hemorrhage whenever the opponent is in range. This works better when some armor is obtained to negate minion retaliation.
Against less skilled opponents, a surprise Apprehend followed by Crippling Strike can force an enemy to tank some turret shots.
Crippling Strike resets your autoattack timer, so use it right after a basic attack to add some extra damage. This can be used for pushing towers.
In teamfights, make sure to get the killing blow when using Noxian Guillotine to refresh the cooldown of the skill so it is available for use again.
Noxian Guillotine deals more damage based on additional Hemorrhage stacks. Try to activate it on targets with max stacks to deal the full damage of the ability.
You can combo Decimate -> Apprehend -> basic attack -> Crippling Strike to quickly apply 3 stacks of Hemorrhage in a very short time. The slow will also allow you to land further attacks to maximise Noxian Guillotine's damage.
Try to avoid pulling enemies in with Apprehend and then using Decimate, as this deals less damage. If possible Decimate first to deal the bonus damage.
Apprehend is a strong crowd control ability in teamfights, but be careful who you pull. Pulling tanks such as Alistar or Amumu might backfire.
Apprehend can pull people and minons over walls. It is useful for seperating enemies from their team or grabbing blue /red buff from a safer location.
Apprehend is a strong initiation skill, but is much better at keeping people in the fight.
Although some believe that Noxian Guillotine is a gap closer since the visual effect looks like it, it is not. The range indicator only shows you how far you can use the ability. You are not moved in the process.
Decimate can be used to check for stealthed champions like Rengar since its animation will become bloodied when successfully hitting an enemy unit.

Your basic combo should be:

Building armor to negate Darius' damage is not as recommended as against other physical damage dealers, since his ultimate deals true damage, his passive deals magic damage and he has passive precentile armor penetration. As such, building health is a more suitable counter.
Even though the damage of Hemorrhage is initially negligible, do not let Darius stack it. This will cause Noxian Guillotine to deal double damage (if you have 5 stacks), and the cooldown of his Crippling Strike will be lower.
Rumble is perhaps the strongest counter-pick to Darius in the lane; Flamespitter harassment is on-par with Darius' and will force him away from the creeps and his farming. Perhaps more importantly, Scrap Shield's damage negating and Electro-Harpoon's slow and range allow Rumble to win trades.
Darius has a high damage output, but he lacks any form of escape mechanism. This makes Darius very prone to crowd control spells.
Remember that Darius has poor sustain. Try to harass him whenever you can to counter him from zoning you.
Remember that towers will automatically target any enemy champion who damages an allied champion within their range. This is important for countering Darius as it means he cannot approach a tower while Hemorrhage is applied to a champion within it's range. Use this to escape Darius by moving near enough to the tower to deny him a kill, or use it to attack Darius via tossing or pulling him to the tower.
Quicksilver Sash not only removes hard disables, but also clears a wide variety of debuffs. Once Darius applies five stacks of Hemorrhage and activates Noxian Guillotine, you can activate Quicksilver Sash while he's mid-cast to greatly reduce the incoming damage from the ultimate. This can potentially turn the fight in your favor and deny Darius the ultimate refresh he needs to continue a rampage.
Shields, especially high-health shields like those from Karma's Soul Shield can be used to deny Darius's Noxian Guillotine a finishing blow, setting it on full cooldown and drastically reducing Darius's presence in team fights
Keep a close eye on low-health allies and/or allies with lots of Hemorrhage stacks, since those are likely to be the targets of Noxian Guillotine.
Shields/heals can be cast during the start up animation of Noxian Guillotine, since the casting time is long enough, and activating a shield too early can result in Darius simply waiting for the bonus health to expire.
Kayle's Intervention can be used to make teammates or yourself invulnerable if you see Darius leaping for Noxian Guillotine.
Abilities that make the user untargetable, such as Fiora's Blade Waltz and Vladimir's Sanguine Pool can be used to nullify Noxian Guillotine. A well timed Sivir's Spell Shield and Nocturne's Shroud of Darkness can also nullify Darius' Ultimate.
Darius is exceptionally vulnerable to being kited, due to the cooldown and short range of Apprehend(the range is shorter in practice than the attack range of every ranged champion with the carry tag), and his lack of other gap closers.
Any sort of champion that can close the gap almost instantly on Darius (for example Jax or Fiora) is also acceptable because of the fact Decimate deals "less" damage inside its range.