Many RP players still don't know how to play their Death Knights.
Here you can find info on them that would fit the lore better and answer some questions.
But before those questions are answered you must know who Arthas is.
"Arthas Menethil, Crown Prince of Lordaeron and Knight of the Silver Hand, was the son of King Terenas Menethil II and heir to the throne. He was trained as a Paladin by Uther the Lightbringer, and had a romantic relationship with the sorceress Jaina Proudmoore. But despite his promising beginning, Arthas became one of the most powerful and evil beings Azeroth would ever experience when he merged with the Lich King." "The Lich King was an entity created by Kil'jaeden the Deceiver from the spirit of the orcish shaman Ner'zhul. Initially trapped within the Frozen Throne, the Lich King eventually merged with the human Arthas Menethil and they became the new Lich King. He is the creator of the Scourge and rules over it telepathically from his throne atop the Icecrown Glacier."
You also need to know that Arthas wouldn't bother raising a dead soldier to be a Death Knight unless he was a hero of their own race. He wouldn't have to be that important, but he wouldn't be a regular Joe.
The best choice, that would fit the lore better, is to be either a Silvermoon Guard or a Blood Knight, in the case of Blood Elves, which most people have.
If you chose to be a Blood Knight, you probably died recently (within 6-8 months or sooner), maybe in one of the battles against the Lich King's Scourge around the cities (the WotLK events before the Expansion). Keep in mind that the time that those attacks occurred is longer than what you actually notice in the game. Those attacks against the cities took months, many died, the cities had zombies and dead people all around. People would be screaming on the streets of the cities, or hiding in the forest with their families, etc. Chaos. During that battle, you could be a Blood Knight and fight against the Scourge, afraid that history might repeat itself, that you would lose family members again, and friends, that everything you own and love would be destroyed (Scourge and Arthas in Quel'Thalas during the Third War). Blood Elves would be terrified by this because they experienced it first hand.
And also, since most of people who play Blood Elves don't realize this... Blood Elves can't understand how Sylvanas is in Undercity with those disgusting savage undead, who were the former Scourge that she died trying to kill. They hate the undead, so they hate the Forsaken, they don't trust them, and think that pretty much everyone else (the other races) are beneath them, because they are the Highborne.
If you chose to die as a hero of your race, in this case the Blood Elves, during the time when Arthas created the Dead Scar through the middle of Quel'Thalas, when Sylvanas was killed and raised as an undead by him, right before he destroyed the Sunwell to bring back Kel'Thuzad; you have to keep in mind that they were still High Elves at that time, the Highborne. Blood elves "renamed themselves in honor of their people that were killed during the siege of Quel'Thalas by the Scourge during the Third War." Before that, there were no Blood Knights, but Silvermoon Guards, and they weren't a lot paladins yet (if any). As high elves, many were members of the Church of Light. Many later became priests and a few became paladins. After the Third War, the majority of high elves became blood elves, and soon the race began to lose its arcane powers (due to the destruction of the Sunwell). Thus they needed to find a new source of power. They did this by sapping Light energy from the captive naaru M'uru held beneath the Blood Knight headquarters.
So, after you chose who you were before you died, you can think about who you are now as a Death Knight.
What is a Death Knight?
They are a "malevolent, rune-wielding warrior of the Scourge" created by the Lich King.
They are UNDEAD (they died and were raised as undead). So, they are immortal.
They still have their soul, knowledge, memory, and personality of the hero they were.
They don't possess free will, The Lich King takes control of their souls and minds, corrupting it, like he did to Arthas.
They only regain free will over their minds after the Death-Knight-specific quest event involving the epic battle against the legendary paladin, Tirion Fordring, the cleansed Ashbringer, and the forces of the Argent Dawn, at Light's Hope Chapel. Then they join the Kights of the Ebon Blade and rebel against the Scourge and the Lich King/Arthas.
"Death Knights are exceptionally skilled in the arts of physical combat and further augment their formidable melee abilities with the use of powerful blood, frost, and unholy disciplines."
How do they become a Death Knight?
There are a few ways that can make sense in the lore, here are some ideas:
1 - They pledged their souls freely into the Lich King's service like Baron Rivendare. In that case you could use the fact that they were promised power and immortal life. That type of character would be evil (and everything else that makes a good cunning personality) even in his mortal life, previous to being a Death Knight.
2 - They were forced into eternal servitude by the Lich King. Killed in battle, thrown on a meat wagon, brought to Arthas and raised as undead. Then they were trained as a Death Knight, with no free will, and having their souls corrupted and tortured by the Lich King. Now serving evil, but still remembering who they were, and the people in their "previous life", they can't go back, they are still evil and will remain for a long time, due to all the corruption and torture to their minds and souls.
How do they think?
On both cases above, Death Knights are evil, but still remember who they were. They want one thing: to destroy Arthas.
They formed the Knights of the Ebon Blade after regaining their free will from the Lich King's control and they want revenge for being betrayed and used by Arthas/Lich King. They can't go back to the person they were before becoming an undead. They don't feel pain, regret, remorse, or even care about anything of anyone.
Making a Death Knight try to be good because they remember who they were is pushing a little, because they can't feel anymore. (It's a similar situation to the Forsaken)
Are they accepted by the Horde/Alliance? Do they accept the Horde/Alliance? Yes, but because they have to. Their faction and races take them back because the Knights of the Ebon Blade associate with the Horde and the Alliance to kill Arthas. They will never be accepted completely by their former races and by the factions because they are seen as traitors, but because the race's leaders allied with the Knights of the Ebon Blade. It is kind of "the enemy of my enemy is my friend" situation.
In RP, if you have to think how people would react around you as a Death Knight... Like in the case of the Blood Elves, once you step in Silvermoon they would look away, close their windows as you pass, frown at you when you look at them. At the Inns and bars they might choose to no serve you, people would leave their seats because you sat close to them etc. You would be hated for being an undead, former Scourge, follower of the Lich King/Arthas, that cost the Blood Elves their city once, and they loved ones.
But you as a Death Knight wouldn't like them that much either. A Death Knight with his free will back is usually still evil, but has to put up with it because if you don't, who would take you? and if you don't have allies, how would you destroy the Arthas? So you pretty much suck up to the Horde/Alliance because that is the only way you can get close enough to kill Arthas.