Lautrec is Evil. His sole aim (that we're aware of) is to kill the Fire Keeper at Firelink Shrine. He may have further goals in mind, but from his dialogue all we can gleam is that he's a pretty nasty character.
As for his staged "meetings", there are multiple ways you can deal with him.
You find him in the Church, And you can do two things: free him, or leave him.
If you leave him, after the Gargoyle boss fight, he will appear at Firelink Shrine regardless. He will not reward you for anything (having done nothing worthy of a reward).
If you free him however, if you travel back down to Firelink Shrine, and speak to him there he will give you a Sunlight Medal. If you do so before the Gargoyle boss fight, you can summon him for this bossfight, as well as the Gaping Dragon boss fight.
To kill, or not to kill..?
From here, there are two ways of dealing with him in future.
Do nothing. After a few interactions, he will offer to sell you some advice (as seen in the dialogues), but little else. Once the player rings both Bells of Awakening, you will find that Lautrec has murdered the Fire Keeper Anastacia, and disappeared. You will be given some items to assist in the revival of the bonfire (If you don't follow this storyline, Firelink will remain dead until the next ng).
Alternatively, you can kill him. I would suggest doing this after the Gaping Dragon fight (as you can summon him to assist), but before ringing the second Bell of Awakening. Once you do, he will drop some Humanity, and the Ring of Favor and Protection.
Pro-tip (Courtesy of GnomeSlice): all you need to do to kill Lautrec in this location is simply kick him two or three times to push him off the ledge where he is currently sitting. Then, reload the area (by quitting or traveling to a new area and then returning) to gain his loot! This is actually a very common tactic in speedruns.
- Note: this ring is quite OP at earlier levels, and most players deem the ring more important than Lautrec's usefulness in the game, and pretty much just kill him at the first available opportunity.
If you chose not to kill him before ringing both Bells, as described above, he will kill Anastacia, and you will receive some items to go fight him later in the game (near the Ornstein/Smough fight in Anor Londo). If the player chooses to fight him here, they will have to invade Lautrec (using the Black Eye Orb), and face off against Lautrec and two other summons. After defeating him here, the player will be rewarded with the Ring of Favor and Protection, Humanity, and a Fire Keeper Soul. Additionally, after the Ornstein/Smough fight, a corpse will appear on the top floor at the back of the boss arena, from which you can obtain his armour set.
A few things to note about this occurrence
The invasion opportunity is only available before the O/S boss fight. If you attempt to trigger the invasion after, it will not be available.
The best way to fight is Ranged. From the spawn point in the hall, if you combine the Hawk Ring with the Dragonslayer Greatbow or Poison Arrows, this fight will go by fairly easily, with no risk to yourself. (At a range, none of the npc's will aggro).
Pro-Tip: You can also use standard arrows to line up your shot first, so you're not wasting valuable poison arrows.
The Fire Keeper Soul that you gain from killing him must be preserved if you wish to revive the Firelink bonfire. Only this Soul can be used to revive Anastacia, and no others. There is no way to identify this soul from any others you might have, so until you choose to revive her, don't use any until she has been revived!