Raynor is a one of a kind as he have a very specific set of skills that makes him an awesome split pusher.
For starters, he have Raiders Recruitment that should help him have mercs on the lane most of the time. He also buffs those mercs with Inspire and Mercenary Lord giving them 62.5% added dmg and 12.5% added atk speed. To make him even more deadly to structures, he can use his Hyperion to bomb them for a lot of damage and hopefully get it up quickly with the help of A Card to Play.
With this build Raynor should not be picking fights, unless it is a team fight. He will usually be dominating the mercenaries while pushing a distant lane and running away, with the help of Confident Aim and Berserk while leaving mercs and Hyperion to destroy the enemy structures.

During the lane phase Sexy Sieger can just stay on a safe distance while eventually hitting an enemy with his Penetrating Round. It is a good idea to get to the sides of the minions in a way that Penetrating Round hits at least 3 of them. If the enemy jumps in to you, use Inspire to raise your minions damage and Penetrating Rounds him away from you while kiting.
Your goal at the start is keep yourself alive to stack Raiders Recruitment the most. You should have one charge, or be real close to have one, when the mercs spawn the first time.
When that happens you should call an ally, recruit one of the mercs and quickly dispose of the other, putting some pressure on your lane. It is a good idea to check the enemy camp if they are not on the lane. If they don't have a Recruiter or Briber they could be in trouble and you should take advantage of that. If they are on lane just rush there and take full advantage of your mercs. When you get the Mercenary Lord it will be key to keep mercs on the lane by your side so keep stacking Raiders Recruitment and whenever you get mercs, stay by their side on the lane. Just remember not to pick fights alone. Your goal here is to keep the pressure while you keep yourself safe and away from the enemy.

When you get your Hyperion you are ready to wreak some havoc. To make a good split push you should go through a far away lane (like bot or top). So get 1 or 2 merc camps for that lane quickly with Raiders Recruitment and head to the lane. Keep the mercs buffed with your Inspire (if you have Steel Resolve it should be up almost all the time) and use your Penetrating Round to quickly clear the enemy minions.

Do not use Berserk to attack the minions or structures!!!

When you get near the structure just throw your Hyperion and keep hitting/clearing minions. Keep an eye for enemy ganks. If any comes to you, even if it is only one, you should leave. Just use your reduced cooldown Penetrating Round knock and Berserk movement bonus to leave the area. Go back to a lane with minions, stack your Raiders Recruitment and repeat.

It is safe to say that you are far from a full dps Raynor when we are talking about team fights. But that doesn't mean that you can't help your team.
First of all, don't be afraid to use Hyperion when the enemy team engages as they usually won't have means to reposition themselves away from the hits.
Secondly you can pop Berserk and keep using your normal attack. Won't hit as much but its a good boost to the team dps. Also, use your Penetrating Round wisely to help team members and disrupt enemy casting and positioning.
If you find yourself being focused just run away while spamming Penetrating Round backwards. Should give you enough time to run or cost the other team too much to chase you.

The talent set is made to give you the split push ability with merc control, merc buff and Hyperion reduced cooldown. But it can be changed for specific situations and team compositions.
The key talents that should not be changed for the split pusher are:
Raiders Recruitment
Mercenary Lord
Hyperion
Steel Resolve

For a more dps approach you could use:
Focused Attack
Berserk
Nexus Frenzy

But it is very risky as it have limited escape options. You will help a lot more on team fight though.

For more mobility you could use:
Bolt of the Storm

To have a even more safe escape.