As stated by Jeremy Crawford in this sage advice answer,
Casting a spell requires all its components unless a trait or feature says otherwise.
Since this feature doesn't specify that you don't need to provide components, you will by RAW have to not only provide the diamond, but also be able to provide Verbal and Somatic components to cast Raise Dead.
Temporary undead
A wizard with Spell Focus (Necromancy) can take the Skeleton Summoner feat, which adds a Human Skeleton (1 HD) to Summon Monster I's list and Human Skeleton Champion (3 HD) to Summon Monster III's list. It also allows you to apply the skeleton template to a summoned creature once per day.
The Zombie Skin Shield allows you to create a zombie for 5 minutes per day, as if you had created them. The price of 2,159 gp may be too much depending on your level though.
Permanent undead
A wizard with the Undead Master archetype can gain the control of undead creatures as if he was a cleric with the Command Undead feat, by spending a few uses of his school's ability. This will not create undead though, only control existing undead, but once controller, they are yours to command.
A tiefling with the right heritage (and some GM's discretion) could be allowed to raise 1 HD zombies/skeletons once per day as a spell like ability.
With Craft Wondrous Items, you could create a Death's Head Talisman. It has a requirement of being able to cast animate dead, which can be bypassed by increasing the spellcraft DC by 5, for a total DC of 15. The item's description says that the creation of the undead is part of the process to create the item, so no onyxes are necessary for that. The item should be affordable aswell by 2nd or 3rd level (625 gp).
If your GM allows 3rd party products, the Deep Magic has a spell called Raise Lesser Undead (2nd for cleric and wizards), which raises a single zombie or skeleton with HD equal to half your caster level (maximum 10th caster level, or 5 HD). It still costs an onyx of 25 gp per HD of the undead created, and you can only have one undead created by this spell under your control. "It's third party...", yes, it is, but it has paizo developers among the designers, including Jason Bulmahn (paizo's lead designer), if that gives the book any credit with your GM.
Best Answer
No, a wizard cannot add raise dead to his spellbook as the spell is not on the wizard's spell list (unless they're a Theurge; see below). The text states that it doesn't need to be in your spellbook to clarify any confusion on whether or not you need it in your spellbook or not.
With the new UA, the wizard archetype of Theurge has access to Raise Dead and can place it in his spellbook at the appropriate level.