Roleplaying alignment has always been subjective and something of a contention point among tabletop gamers. How you treat alignment during combat should be a natural extension of how you treat it outside of combat. Here are a few guiding questions I usually use for my characters.
How does my alignment affect my relationship with the party?
How you view your allies is going to depend on your alignment and character type. A Lawful Good Paladin might go out of the way to ensure the party acts nobly and with honor. In combat this character usually plays team leader, since he is always so sure his way is the right way.
A Neutral Good Cleric could see the party as a literal godsend, a way for them to further their chosen deity's goals and help the world. During combat they would be more concerned for the party's well being, always ready to lend a healing or buffing spell to an ally in need.
In my mind, Neutral Evil is the most selfish alignment. So your character could view his party as mere tools to further his own agenda. You don't discard a good tool for no reason, so why would you act in a way that made the party leave/turn on you? That said once a tool has outlived its usefulness to you feel free to leave it behind.
How does my alignment affect my relationship with the enemy?
Like above, how you handle conflict can be very alignment driven. That Lawful Good Paladin might advocate taking captives alive so they could be put on trial, rather than just outright slaughtering villains. He'll call for and accept an opponents surrender, and expect the party to honor his agreements.
The Neutral Good Cleric doesn't want anyone to suffer and is always looking for converts, so she might tend to wounded enemies or preach to captives. If there is a non-violent, or at least non-lethal, way of handling a situation then that will be her preferred option.
As for your character, favoring strength over weakness means that during fights you would be aiming to kill your enemies to prove your superiority over them. If you have a character in the party like the above Paladin who wants to take prisoners you can accede to his wishes as long as you need him around. And if you decide to threaten/extort the prisoners during one of those long, lonely night shifts then that is nobody's business but your own. Similarly, anyone who is stupid or weak enough to be kidnapped or taken hostage deserves their fate. As long as you can convince the party that any collateral damage was an accident or unavoidable then who cares what happens to them.
@Neil Slater has a lot of good suggestions in his answer on what to focus on for an evil alignment. The one I would most emphasize is make sure your character has a goal. This is important for roleplaying any character, but doubly so for a character who may have different morals than the rest of their party. Even if your reason for working with them is as simple as "I can't take over the world if the Big Bad destroys it", that is at least enough to justify your character's behavior and actions. Then you could say things like "Little Timmy deserves to die for falling in that well, but if I save him the Paladin will trust me more. That will make stabbing him in the back later easier." Son ow you could even perform "good" actions as long as they further your, ultimately evil, long-term goals.
The important thing to remember is that the stereotypical Evil Overlord should not be your role model. They should serve as a warning about the mistakes an effective Evil character shouldn't make.