Currently experiencing this same problem. Earned First Blood in the Crucible yesterday (proof in the link under medals earned) and no achievement 24 hours later. As someone who doesn't play crucible very often its kind of a rare occurrence for me to get first blood.
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Go to Destiny's Settings > Log Out > Switch Profile. Choose your existing Xbox profile to login again. Once you choose your/a character and get back to Orbit, you should get whatever achievement(s) you should have gotten before.