No, because the mount acts on a separate turn from yours

Charger says:

When you use your action to Dash, you can use a bonus action to make one melee weapon attack or to shove a creature.

• If you move at least 10 feet in a straight line immediately before taking this bonus action, you either gain a +5 bonus to the attack's damage roll (if you chose to make a melee attack and hit) or push the target up to 10 feet away from you (if you chose to shove and you
  succeed).

Step 1

So the basic idea of your step 1 is perfectly fine. You take the dash action, which is allowed and this means that your movement increases. However, do note that this doesn't affect your mount at all (so no "increasing speed by 10 ft" for example). Also it is good to note here that dash does not modify any speed at all; it simply grants additional movement.

Step 2

The problems start around step 2 however.

Since you have used the dash action, you may now take a bonus action attack against any enemy in range. However, in order to get the +5 bonus damage as well you must first move at least 10 feet in a straight line immediately before taking this bonus attack.

You might think that moving your mount would work, but it does not. This is because a controlled mount seems according to the rules to act on a turn separate from yours. Per Jeremy Crawford:

A rider and a controlled mount have separate turns, but they have the same initiative, which means you decide which one goes first.

Thus, you cannot intersperse your and its movement and actions (see this answer for more details on this and the debate surrounding it).

This means that you must end your turn in order to have the mount move towards the opponent. Since it is no longer your turn, you cannot take the bonus attack (bonus actions must be taken on your turn) or the +5 damage (because it is no longer immediately after your dash action).

However, if you dashed then dismounted and moved 10+ feet in a straight line towards an enemy you would definitely be able to get your bonus attack with +5 damage.

In general, riders and mounts can be targeted like any other creature

The rules for Mounted Combat don't specify how targeting works for the Attack action (or any other action) because targeting riders and mounts is no different that targeting any other creature, with the exception of opportunity attacks.

Whether you can target either, neither, just the rider, or just the mount has nothing to do with their status as rider and mount, rather it depends on many other circumstantial factors, the most common of which are by far range and total cover.

From Making an Attack:

Choose a target. Pick a target within your attack's range: a creature, an object, or a location.

From Total Cover:

A target with total cover can't be targeted directly by an attack or a spell, although some spells can reach such a target by including it in an area of effect. A target has total cover if it is completely concealed by an obstacle.

For example:

• If your attack has a 5' range and you are a medium creature standing at the feet of a T-Rex, then you can target the T-Rex, but you probably¹ can't target its rider.
• If you can see a rider peeking over a wall, but its mount has total cover, then you can target the rider with an attack, but not the mount.

Opportunity attacks

The rules for Opportunity Attacks state that:

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

And that:

You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

The rules for Controlling a Mount also state that:

if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.

In short, a rider never provokes your opportunity attack when it moves using its mount's movement. But if a mount provokes your opportunity attack, then you can choose to target the rider instead of the mount, despite the fact that the rider did not provoke said opportunity attack.

However, this does not mean you can target a rider regardless of all other factors such as range or total cover. For example, opportunity attacks occur retroactively just before the target leaves your reach, but if the rider was never within your reach to begin with, then you can't possibly attack the rider.

1. The height of creatures and the location of a rider atop a mount are often not well defined so there is room for DM adjudication.