If I cast dimension door can I also teleport myself onto a higher vertical plane in the air?
For instance, if I have the feat "Death from above" can I use my teleport to gain a high ground requirement?
The spell dimension door allows the caster to travel instantaneously "from [the caster's] current location to any other spot within range. [Casters] always arrive at exactly the spot desired—whether by simply visualizing the area or by stating direction." Up—to the limit of the spell's range and the caster's desires—is certainly possible (albeit risky if the destination hasn't been scouted first).
The spells teleport et al. can do the same, although there's a chance of mishap with some effects and the caster "must have some clear idea of the location and layout of the destination." Achieving knowledge of up by looking skyward is totally legit.
However, it's really complicated to employ an instantaneous travel effect like those above so that the benefit of the feat Death from Above can be realized as that feat's benefit requires the creature to "charge an opponent from higher ground… [to] gain a +5 bonus on attack rolls in place of the bonuses from charging and being on higher ground."
That is, while making the charge the creature must take a free or swift action to activate the instantaneous travel effect then the GM must rule that the creature did not violate the stiff restrictions on making a charge in the first place: a charging creature "must move to the closest space from which [it] can attack the opponent." Since taking a full-round action to charge is nearly unique among actions in that it requires a declaration of intent before the action's actually taken, a GM may rule that the space is picked when the charge is declared and that the creature's charge is foiled by its own mid-charge instantaneous travel effect, probably because the creature ends up attacking from a different space than one the creature would've been attacking from when the charge was declared. Ask the GM.
However, keep in mind that unless the creature can also fly, after the creature's instantaneously traveled upward, the creature will fall and—unless the surface is more than 500 ft. below the creature—, also, land. Like, right now! There's no chance in such cases, for example, for the charger-now-caster-now-charger-again to attack the foe from higher ground before falling, cartoon physics to the contrary.
Dimension door, teleport, and rules for conjuration (teleportation) spells do not mention any restriction for teleporting subjects onto solid ground. This is probably an intentional omission since other spells (like summon monster) do have such restrictions. This makes sense as you might have to teleport into the air when plane shifting to places that have no solid ground.
However, action economy may prevent you from using the Death From Above feat since that feat requires you to perform a charge action while having the high ground.
From the description of dimension door:
After using this spell, you can’t take any other actions until your next turn.
In particular, you can't attack, nor attempt to stabilize your flight. You will be able to do that on the next round, if somehow you are still in the air.