[Update Jan 2022] A much better solution with a lot of fixes and enhancements was created by the community and is available here: https://github.com/rr-/Tomb1Main
I will rewrite this guide once I have enough time. Meanwhile, you can apply this guide and then download Tomb1Main on top if it.

Tomb Raider was released for DOS in 1996. Several patches were made to support the different video cards available back then. The last patch released was for ATI cards that supported the ATICIF API (think something similar to OpenGL or DirectX). Unlike previous patches, this one was a Win32 exe, compatible with Windows 95. Unfortunately, the patch was forgotten, due to the lack of modern ATICIF hardware and incompatibility with newer versions of Windows.

Enter barracuda415, who made glrage[github.com], an OpenGL wrapper for ATI3DCIF, patched the game and added support for a bunch of other features.

This guide will help you run Tomb Raider natively on Windows without Dosbox getting in the way. The result is an easily configurable and moddable game.

Note: If you want a faster and easier way to do this, use the glrage version provided by Glidos:
http://www.glidos.net/tombati.html
It's an installer that does most of the work for you, and provides extra features like texture packs and high res videos. If you're having problems with the Glidos version and want to try vanilla glrage, continue reading.

• Windows 7 or higher
  
• An OpenGL 3.3+ compatible graphics card (anything made after 2012 should work)

We are going to rebuild the game folder.

1. Browse to your local game files. The path should be
   "...\Steam\steamapps\common\Tomb Raider (I)"
2. Delete all folders and files except GAME.GOG and GAME.DAT
   
3. Open GAME.DAT with notepad, change the first line to
   FILE "GAME.BIN" BINARY
4. Rename GAME.GOG to GAME.BIN, and rename GAME.DAT to GAME.CUE
   
5. Mount and open GAME.CUE with your favorite mounting tool (I use WinCDEmu)
   
6. Copy the "DATA" and "FMV" folders to "...\Steam\steamapps\common\Tomb Raider (I)"
   
7. Eject the CD image, delete GAME.BIN and GAME.CUE
   
   
8. Download this archive:
   
   mediafire .com/file/q8eeap6756a95sp/TOMB1.zip (remove the space)
   This contains the ATI patch, glrage, the music files, and audio wrapper. All preconfigured and ready for Steam. Extract to "...\Steam\steamapps\common\Tomb Raider (I)"
   
   Your Tomb Raider (I) folder should now look like this:
   
   
   
9. Download and install Visual C++ Redistributable for Visual Studio 2015[www.microsoft.com] (both x86 AND x64)

The game is now ready to be played through Steam.

• You can use JoyToKey[joytokey.net] or Xpadder[www.xpadder.com] to map your controller buttons. If you can't map the cursor keys, try mapping the numpad cursor keys instead, or configure the game to use WASD for example and then map the new keys to JoyToKey.
  
• If you notice screen tearing, force Vsync on from your graphics card control panel. (The vsync option inside glrage.ini doesn't have any effect, at least on my system). You can also experiment with forcing triple buffering on or off.
  
• You can force Anti-aliasing in your graphics card's control panel. Super sampling gave me the best result. Black lines appeared for me when using other modes. Experiment to get the result you want.
  
• Use ALT+Enter to switch between full screen and windowed mode.
  
• You can force the game to always run in windowed mode. Open glrage.ini and set
  fullscreen_mode = 2
• If you have a multi-monitor setup and want the game to display on all monitors instead of the primary one, open glrage.ini and set
  fullscreen_virtual = true
• In dosbox.ini, you can set a resolution higher than your native one. This reduces polygon wobble.
  
• In dosbox.ini, you can increase Draw Distance by setting
  draw_distance_override = true
  This might change the original mood.
  
  
• In dosbox.ini, you can enable PSX-like turquoise water color by setting
  watercolor_override = true
  

1. Download the game files
   mediafire .com/file/9j638b38g35ifhp/TRUB.zip (remove the space)
   This contains the game files preconfigured with glrage and ready to play. Extract anywhere you want outside of the Tomb Raider (I) folder.
   
2. If you haven't done so already, download and install Visual C++ Redistributable for Visual Studio 2015[www.microsoft.com] (both x86 AND x64)
   
3. Add Tombub.exe to your Steam library.

All the optional configuration above can be applied to TRUB, but instead of editing dosbox.ini, you should edit tombub.ini.

ata4 (barracuda415) for making glrage, fixing the ATI patch, audio solution and adding features and enhancements.