I believe you need to play through the entire game without shooting anyone to gain this achievement. If you mess up, you can reload your last checkpoint and go again but if you've passed another checkpoint since, you'll need to restart.

If you want to check as you go along, you can press the "Back" button and in the top-right of the screen it should say: "So far, you have not fired a shot at anyone". If you think you may have shot someone, hit the "Back" button and check this. If it's gone, you need to reload your last checkpoint. If it's there, you're good.

The bit with the convoy doesn't count as a shot and you can actually shoot guns in the game (at servers etc), as long as you don't shoot at anyone.

It seems that Chance has found the solution for his crash. I figured that this may be a good opportunity to make a general Fallout: New Vegas crash/bug troubleshooting list for everyone here. It's Community Wiki (notably because most of the content comes from The Vault, so I can hardly be credited), so please feel free to edit and add more to this.

Some of these issues may be resolved in more recent patches; if so, please update this answer and remove them.

All Platforms

• Always use the latest patch of Fallout: New Vegas
• Always use multiple save slots -- a lot of issues in Gamebryo-based games come from corrupted save files
• Do not rely on Autosave; save manually, save early, save often
• Crashes often when you near completion of the game.
• Crashes in VATS are somewhat common.
• Crashing or freezing when entering or exiting areas is somewhat common.
• Crashing or freezing upon entering The Strip is fairly common.
• Crashes or very severe slowdowns are common when the player character reaches the top of a hill or ridge and the landscape and sky beyond become visible.
• If one saves at the beginning of the game while editing your character the Level up system can crash.
• As for any game, crashes may mean defective hardware.

PC

(Note: I have zero experience with the PS3 version; all of this comes from The Vault or my own experience troubleshooting other PC games and issues.)

• Make sure your system meets (or preferably exceeds) the minimum requirements to run Fallout: New Vegas
• Keep all your system drivers (audio and video in particular) up-to-date
• Do not ALT+TAB out of Fallout: New Vegas.
• If you have installed custom mods, try removing them and seeing if the issue disappears
• Scrolling the list of saved games can cause the interface to freeze and crash.
• If you're still experiencing crashes, head over to Bethesda's PC Issues page and see if others are experiencing them, and/or if a fix is known.

X-Box 360

• Install the game if you have the hard drive space available for it. (This has helped me on every glitchy/slow-loading game I play.)
• If, after installing, you get messages that the disk is unreadable, uninstall and then reinstall the game -- the install files are likely corrupt.
• Don't equip cowboy hats via a mapped key (that is, the D-Pad shortcuts). Why you'd want to is beyond me, but I guess someone did it and found at that it can occasionally make the console freeze up. If you want to equip a cowboy hat, take the extra minute or so and do it through your PIP-Boy.
• Slightly contrary to the advice under "All Platforms," having too many saved games can cause crashes. (I hope that means a truly excessive number, and not the dozen or so I have ... )
• Entering Dead Sea's Barracks in Nelson may cause the game to freeze. This may have been caused by meeting Dead Sea in Nipton beforehand.
• The game may freeze when repairing/wearing/unequipping the NCR Bandoleer Armor found in the tent next to Lieutenant Hayes' Tent and in the Supply Shack at Camp McCarran.
• Equipping the sniper rifle may cause the game to slow down severely and most likely freeze.
• If you are experiencing random crashes, clear your X-Box 360 System Cache.
• If you are still experiencing crashes, head over to Bethesda's X-Box 360 Issues page.

Playstation 3

(Note: I have zero experience with the PS3 version; all of this comes from The Vault.)

• The game may freeze when repairing/wearing/unequipping the NCR Bandoleer Armor found in the tent next to Lieutenant Hayes' Tent and in the Supply Shack at Camp McCarran.
• Equipping the sniper rifle may cause the game to slow down severely and most likely freeze. On the PS3 equipping the sniper rifle can cause the weapon hotkeys to stop functioning, and the game to crash shortly thereafter on attempts to sleep, save, enter a new area, or just walk around the map for a minute or so.
• Sometimes changing the color of the interface can cause the game to crash or lock up. Keeping the interface Amber may help in avoiding this. (On both the Pip-Boy and Hud)
• If you're experiencing random crashes, check Bethesda's PS3 Issues page