Is there any way to see progress towards achievements in Civ 5?

This file has been removed as of the January patch. As such, as of now there is no method to check achievement progress.

Actually, there is a way to view achievement progress, although it is a little troublesome to find and interpret. In the folder Documents\My Games\Sid Meier's Civilization 5\Logs is a file called "achievements_debug.log". If you open it, and scroll down you can find the values of different steamstat variables that can be interpreted into your progress for the achievements. For example:
[49597.863] DBG: Steam Stat: ESTEAMSTAT_ROADS = 85
This line says that I have built 85 roads, and the achievement that had to do with roads is to build 1000, so I would have to build 915 more to get the achievements. The file should contain everything you need to know about your progress on the achievements. If you need any help, feel free to ask.

Unless the game itself is programmed to display your progress towards the steam achievements (Team Fortress 2, Left 4 Dead, Alien Swarm, Trine, etc.) there is no way to know how close you are, barring meticulous tracking on your own.

This question dates a little but in my opinion is still very relevant being in the top 10 daily on Steam Stats. GunnarJ has the right idea but now you need to re-enable the debug log:

1. Open the config.ini file (located in Documents\My Games\Sid Meier's Civilization 5) and change LoggingEnabled = 0 to LoggingEnabled = 1.
2. Re-start Civ-V and you should now have the achievements_debug.log file available in Documents\My Games\Sid Meier's Civilization 5\Logs

Here's a video if you prefer.

There's also a tool you can use to analyze the debug file: civ-a-log.com if you don't want to go through the pain of analyzing it yourself.