How can I kill vampires?

Vampires are really tough to fight against, especially the master vampire variants as they can take a huge beating. However, all vampires are weak to fire (they take an extra 50% damage). What you want to do ahead of time is to prepare lots and lots of fire: cast flame cloak, put a fire rune on the ground, and summon a flame atronach. I also noticed lots of vampires prefer frost type spells, so having a potion or two of frost resistance may help.

Some alternative tactics:

• Lure them out into the open (assuming its daylight). The sun will weaken them, reducing their health, stamina and magicka by a certain amount (but not kill them)
• Take advantage of dungeon traps, use those to take out vampires.

There are silver weapons that appear when fighting the silver hand those weapons are pretty good against them as well as fire.

The question predates the release of Dawnguard, so I will assume no add-ons are installed.

Vampires are weak against fire. Fire-based Destruction spells are the most direct way to exploit this, but since your character is designed primarily for Conjuration, I'd probably recommend Conjure Dremora Lord, assuming your character can cast it. If you have the Daedric artifact Sanguine Rose, you can use that as well. Both will conjure a Dremora equipped with a fire-enchanted Daedric greatsword. You can also look into putting a fire enchantment on your weapon to inflict added damage.

Speaking of Daedric weapons, it looks like your character could be using one. Your question states that your character has Dragonplate armor and an Orcish sword both tempered to Legendary. If your character is able to make Dragonplate armor (and I'm assuming you took the right-hand branch on the Smithing perk tree given the tempering on your Orcish sword), your character should be able to make Daedric gear. If you don't have Daedra hearts, you can still make Ebony weapons that are noticeably better than the low-level Orcish weapons. (Dawnguard adds Dragonbone weapons that can be made in the same fashion as Dragonscale and Dragonplate armor and outperform even Daedric weapons.)