To use any controller with the early series of Tomb Raider games, you need to use a "key mapper" or "emulator" program which makes the controller output produce keyboard inputs. TR-I runs through dosbox which has a built-in key mapper, but it is not easy to use. The other TR games do not detect all buttons on some controllers (such as D-pad and triggers).

I recommend using a separate program such as JoyToKey - it works on Windows with both XInput and DirectInput controllers, and you can re-use profiles with later TR games (with just some minor adjustments). It can also emulate mouse function.
Other programs such as Xpadder, Pinnacle Game Profiler, Logitech Profiler, or Steam's in-built controller configuration (Steam Input) provide similar functions. Make sure you are only using one key mapper at a time - for example, you may need to disable Steam's controller configuration for JoyToKey to work as expected.
To use Steam Input with the old TR games, you must disable or fix the Steam Overlay:
https://psteamcommunity.yuanyoumao.com/app/225300/discussions/0/1639793203775973733/

This guide assumes that you already have the necessary USB connectors, wireless receivers, drivers, etc, for your gamepad/controller to be functional on your computer. Note that a "play & charge" kit for a wireless Xbox controller does NOT "connect" it to your computer.

For each game in the TR series, read the instruction Manual for explanations of how all the controls work. A link to an electronic copy of the manual (in .pdf format) can be found in the Additional Content section of the game's Steam Library page.

Note that controls for TR games may have a set of "Default Keys" that cannot be changed, and a set of adjustable "User Keys", and they are BOTH ACTIVE at the same time, so be careful not to end up with two actions for the same key or button.

An excellent resource for all Tomb Raider games is Stella's Tomb Raider Site:
http://tombraiders.net/

Download and run JoyToKey (current Ver 6.3): https://joytokey.net/en/
(If you run the game in a compatibility mode that requires administrator privilege, then JoyToKey may need to be "Run As Administrator".)

To create a profile, press the "Create" button, and enter a name.
In the Options tab, for "Stick/POV" select "Stick x 2 + POV x 1",
and set "Threshold for input" (for analog sticks) to ~30%.

In the Joystick 1 tab, assign the controller buttons to default keyboard keys:
1. Press any button on the controller to highlight it in yellow in the list.
2. Double-click the control to open it for editing.
3. In the highlighted field, press the keyboard key you want to assign, then click OK.
Repeat 1-3 for other controller buttons.
(Example: Press the [A] button on an X360 controller, double-click the highlighted line on the Joystick 1 tab (Button1), press Ctrl on keyboard, then click OK.)
If you like, add a comment to each item describing button name or game action.

Save the profile(s) (File > Save).

For profiles to be automatically selected when you play a particular game, go to the Settings menu and choose "Associate profiles with applications...". Note that Tomb Raider I on Steam starts with the application "dosbox.exe" (including gold expansion and patched versions).

See full example controls and screenshots below.
For the first six games in the Tomb Raider series, you can use three profiles.
Downloadable profiles for Xbox/Xinput controllers:
Tomb Raider JoyToKey Profiles[www.dropbox.com]
(Extract the .cfg files into your \Documents\JoyToKey folder, then restart the JoyToKey app.)

Note: since version 6.1 of JoyToKey, the games are responding to "L-Alt" instead of the generic "Alt" for the jump action, so the profiles have been updated.

The TR games automatically use the left-thumbstick for movement (run/back/turn), which can cause menu "spinning" or unwanted movement due to the thumbstick not staying centered. If you prefer to use the left-thumbstick for movement, it can be set in JoyToKey with a more suitable deadzone (threshold), so it is better to completely disable automatic detection of the joystick by the game.

For the dosbox version of TR-I, open dosboxTR.conf (in the Tomb Raider I folder) with a text editor (eg Notepad). In the [joystick] section, set joysticktype=none and save the file.

For other TR games, either "disable joystick" on the setup screen (launch option: -setup) or only select "keyboard" control method from within the game.

Both thumbsticks will then be available for additional functions via JoyToKey (such as weapon selection, health packs, or save/load shortcuts).

If the Joystick 1 tab in JoyToKey is not highlighted when you press the controller buttons, there may be another device connected to your computer that is being detected as the "primary controller".

The Options tab in JoyToKey will indicate how many joysticks are detected. Try increasing the "Number of joysticks to configure", and see if a different tab is highlighted when you press the controller buttons, then copy the configuration to that joystick.

Another option would be to go to the "Control Panel" on your computer and open the "Devices and Printers" window. Right-click on your controller, select "Game controller settings", click the "Advanced..." button, then select your "Preferred device" from the drop-down list and click OK.

To fix "controller detection" problems, the solutions involve disconnecting other devices or disabling/uninstalling drivers that may be causing conflict with the controller.

A common solution is to disable an unnecessary "HID-compliant game controller" in Device Manager, as described in these other discusions:
http://psteamcommunity.yuanyoumao.com/app/225080/discussions/0/616198900640400935/
http://psteamcommunity.yuanyoumao.com/app/225080/discussions/0/1368380934269406978/

Other examples that may help:
- Disable joystick drivers, such as "Generic virtual HID driver"
(Device Manager > Human Interface Devices > "Joystick name" > Disable)
- Disable bluetooth driver (device manager), or unpair/unplug USB bluetooth receiver for controller
- Disable or uninstall other controller/keyboard mapping utilities (eg Wii U Nintendo Gamecube controller utility, MS Intellitype, Xpadder, etc)
- Disconnect webcam, or some types of keyboard/mouse (mistakenly detected as a controller)
- Uninstall virtual joystick (Vjoy)

Example controls based on PlayStation defaults:

Tomb Raider I
Tomb Raider II (additional controls)
One possible variation from the default controls is to use the triggers for look/walk and the bumpers for sidestep/etc/etc.

For "Jump", it may be necessary to use "L-Alt" instead of "Alt".
Right-click on the entry field in JoyToKey, and select "L-Alt" from the list.

If using a laptop computer without a numeric keypad (Numpad), you will need to reassign the "Look" key from within the game Controls > User Keys. (For example, set it to the letter "O".)

"Copy" and edit the previous profile for Tomb Raider I & II.

Tomb Raider III (modified controls)
Tomb Raider IV & V (additional controls)

"Copy" and edit the previous profile for Tomb Raider III & IV & V.
Default PS2 controls are shown for reference, but the suggested Xbox controls are similar to the previous five games.

Tomb Raider VI
The default "look" function is based on mouse movement, and has a non-inverted y-axis when using the movement keys. To make it similar to the previous games, you could additionally assign movement to one of the thumbsticks with the up/down directions swapped.

If using a laptop computer without a numeric keypad (Numpad), you will need to reassign the "Sprint" and "Duck" keys from within the game Controls > Keyboard. (For example, set them to the letters "O" and "I".)