Character is stuck between grid cell gap

Question

I started making a D&D-style turn based tactic game. I have a problem with my tilemap.

I made a script to find the location on the tilemap for my character and for the tile I clicked with the mouse. (Both are working: it shows me locations like (3,-2) etc.)

But when I try to move my character from location to target... My character is stuck, always in one place ~(0,0) in the gap between cells. Here is the code.

using UnityEngine;
using UnityEngine.Tilemaps;
public class CharacterMovement : MonoBehaviour
{
    public Tilemap tilemap;
    public float moveSpeed = 50f;
    public bool hasMoved = false;
    private Vector2Int targetPosition;
    private Vector2Int characterLocation;
private void Update()
{
    MoveToClickedTile();
}

private void MoveToClickedTile()
{
    // Check for left mouse button click
    if (Input.GetMouseButtonDown(0) && !hasMoved)
    {
        GetMouseLeftClickPosition();
        GetCharacterTileLocation();
        // works
        print(targetPosition + "target");   
        print(characterLocation + "character");

        // doesnt work
        transform.position = Vector2.MoveTowards(characterLocation, targetPosition, moveSpeed * Time.deltaTime);

        hasMoved = true;
    }
}

private void GetMouseLeftClickPosition()
{
    Vector2 mouseWorldPos = Camera.main.ScreenToWorldPoint(Input.mousePosition);
    targetPosition = (Vector2Int)tilemap.WorldToCell(mouseWorldPos);
}

private void GetCharacterTileLocation()
{
    characterLocation = (Vector2Int)tilemap.WorldToCell(transform.position);
}

}

Is the problem that my character does not "belong" to the tilemap? I made it as a separate game object, and its transform location is really strange, I don't understand why.

But when I made the character separate from the tilemap, it had a weird box that was much more bigger than the character, and I could not click on it to find out what it was.