Making a UI.Button put text into a InputField

You don't really need an input field just a text field will do it if you want to make a custom keyboard, this is in C#...

 using UnityEngine;
 using UnityEngine.UI;        // add at the top
 using System.Collections;
 
 public class MyKeyboard : MonoBehaviour {
 
     public Text myText;        // drag your text object on here
     private bool shiftOn = false;
     
     public void ClickLetter(string letterClicked)
     {
         string tempCurString = myText.text;
         
         if(shiftOn)
             letterClicked = letterClicked.ToUpper();
                 
         string tempNewString = tempCurString + letterClicked;
         myText.text = tempNewString;
     }
     
     public void ClickBackspace()
     {
         string tempGetString = myText.text;
         if(tempGetString.Length > 0)
         {
             string tempString = tempGetString.Substring(0, tempGetString.Length -1);
             myText.text = tempString;
         }
     }
     
     public void PressShift()
     {
         shiftOn = !shiftOn;
     }
 }

Drag that onto the panel your custom keyboard is on and add all your buttons, including Shift and Backspace. In the OnClick for each button click + and drag the panel with the script on it onto the slot that appears.

In the OnClick drop down for the letter/number buttons select MyKeyboard -> ClickedLetter and set the string field to be the lowercase letter for that button.

On the Shift/Backspace do the same but select PressShift/ClickBackspace.

Don't forget to drag the Text object onto the Panels Text field in that script.

Should work fine.