The Player Controller script stripped to it's core.

PlayerCharacterController.cs

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class PlayerCharacterController : MonoBehaviour {
	//internal components
	Rigidbody2D rigidBody;
	BoxCollider2D boxCollider;

	//constants
	const float deadZone = 0.25f; //axis input deadzone
	const float groundedProjection = 0.08f; //how far into the ground to project the linecasts

	//gameplay (DOCS: These settings are calibrated for 1 pixel = 1 unity unit)
	[Header("Movement Settings")]
	public float gravityScale = 100;
	public float moveForce = 1000f;
	public float jumpForce = 30000;
	public float maxSpeed = 300f;
	public float fallSpeed = 800f;

	//basic movement
	float horizontalInput = 0f;
	float verticalInput = 0f;
	bool jumping = false;
	bool grounded = false;

	void Awake() {
		rigidBody = GetComponent<Rigidbody2D>();
		boxCollider = GetComponent<BoxCollider2D>();
	}

	void Start() {
		//
	}

	void Update() {
		HandleInput();
	}

	void FixedUpdate() {
		//debug
		rigidBody.gravityScale = gravityScale;

		HandleMovement();
	}

	void HandleInput() {
		//determine if on the ground (using coyote time)
		bool trueGrounded;
		trueGrounded  = Physics2D.Linecast(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x), -groundedProjection, 0), 1 << LayerMask.NameToLayer("Ground"));
		trueGrounded |= Physics2D.Linecast(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x + boxCollider.size.x / 2.1f), -groundedProjection, 0), 1 << LayerMask.NameToLayer("Ground"));
		trueGrounded |= Physics2D.Linecast(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x - boxCollider.size.x / 2.1f), -groundedProjection, 0), 1 << LayerMask.NameToLayer("Ground"));

		if (trueGrounded) {
			grounded = true;
		} else {
			StartCoroutine(SetGroundedAfter(false, 0.1f)); //coyote physics: 100ms
		}

		//get inputs
		verticalInput = GamePad.GetAxis(CAxis.LY);
		horizontalInput = GamePad.GetAxis(CAxis.LX);

		//flip direction
		if (Mathf.Abs(horizontalInput) >= deadZone) {
			transform.localScale = new Vector3(horizontalInput > 0 ? 1 : -1, 1, 1);
		}

		//determine if jumping
		if (GamePad.GetState().Pressed(CButton.A) && grounded) {
			jumping = true;
		}
	}

	void HandleMovement() {
		//stop the player if input in that direction has been removed
		if (horizontalInput * rigidBody.velocity.x <= 0 && grounded) {
			rigidBody.velocity = new Vector2 (rigidBody.velocity.x * 0.85f, rigidBody.velocity.y);
		}

		//move in the inputted direction, if not at max speed
		if (horizontalInput * rigidBody.velocity.x < maxSpeed) {
			rigidBody.AddForce (Vector2.right * horizontalInput * moveForce);
		}

		//slow the player down when it's travelling too fast
		if (Mathf.Abs(rigidBody.velocity.x) > maxSpeed) {
			rigidBody.velocity = new Vector2 (Mathf.Sign(rigidBody.velocity.x) * maxSpeed, rigidBody.velocity.y);
		}

		if (rigidBody.velocity.y < -fallSpeed) {
			rigidBody.velocity = new Vector2 (rigidBody.velocity.x, Mathf.Sign(rigidBody.velocity.y) * fallSpeed);
		}

		//jump up
		if (jumping) {
			rigidBody.velocity = new Vector2(rigidBody.velocity.x, 0f); //max v-jump speed
			rigidBody.AddForce (new Vector2 (0f, jumpForce));
			jumping = false;
		}
	}

	void OnDrawGizmos() {
		if (boxCollider != null) {
			Gizmos.color = Color.red;
			Gizmos.DrawLine(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x), -groundedProjection, 0));
			Gizmos.DrawLine(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x + boxCollider.size.x / 2.1f), -groundedProjection, 0));
			Gizmos.DrawLine(transform.position + new Vector3(boxCollider.offset.x, boxCollider.offset.y, 0), transform.position + new Vector3(transform.localScale.x * (boxCollider.offset.x - boxCollider.size.x / 2.1f), -groundedProjection, 0));
		}
	}

	IEnumerator SetGroundedAfter(bool value, float delay) {
		yield return new WaitForSeconds(delay);
		grounded = value;
	}
}