Ideally for them, the Confederates would have to win First Manassas (July 21, 1861) decisively, thus setting the tone for the Eastern Theatre. If that then proceeds roughly the way it did in OTL up to Second Manassas (August 28-30, 1862), it will very much seem like the Confederates are going strong. This will lead to yet more of the same (overly) cautious hesitation on the Union's part. After that, have butterflied culminate in complete surprise at Battle of Antietam (September 17, 1862) as was intended, leading to a clear Confederate victory. [Obviously, I'm killing butterflies here; assume a similar but not identical set of battles occurring.]
That's the Eastern theatre. In the West, I don't think Shiloh would be enough. Again, ideally for the Confederates, they'd have to dedicate themselves to striking hard, and preventing the OTL Union victories at Fort Henry (February 6, 1862) and Fort Donelson (February 11-16, 1862). This would deny the Union its OTL control of the Tennessee and Cumberland Rivers. No fall of Nashville shortly thereafter. In an analogue to Shiloh (April 6-7, 1862) the Confederates should beat back the Union decisively. Also, A.S. Johnston should survive, since losing him was a big hit. This keeps the Confederates able to hold the line, and (combined with the lack of Union control of the rivers) prevents the fall of Memphis.
Finally, and this may well be the hardest to achieve, the fall of New Orleans must be prevented. Even if you keep the line in the Western Theatre up North, the war is lost as soon as the Union controls the mouth of the Mississippi. Once that happens, you may as well surrender and try to negotiate a deal where you publicly denounce secession as a mistake (and illegal) in return for blanket amnesty-- something Lincoln was more than willing to offer. So you need to ensure Farragut fails to take the city, no matter the cost.
At that point, it's just a matter of waiting it out in the West and South, and keeping up the pressure in the East. The ATL Antietam victory would bolster the Confederates and prevent the Emancipation Proclamation (since Lincoln was waiting for a victory to announce it). With the Confederates being perceived as more dangerous than in OTL, the invasion of the North would look like a success. A Fredericksburg or Chancellorsville analogue on Northern soil would be within reach, and once that is achieved, it becomes the decisive battlefield victory that Lee wished gettysburg to be in OTL. With this string of Confederate successes and Union failures, you may except diplomatic recognition of the Confederacy by Britain and/or France at the close of 1862.
After that, no more stupid risks. Fall back a bit, dig in, prepare your defensive positions and offer whoever recognises you formidably lucrative trade deals... just so long as their ships come pick up the goods at your ports. Because then, Lincoln will presumably be faced with the option of ending the blockade, or firing on European vessels. And that's when he's lost the war. Peace negotiations by summer '63, definitive treaty before the year is out.
To be clear: this demands improbable amounts of luck for the CSA. It would require the Confederates to win six more battles than in OTL (First Manassas, Antietam, Fort Henry, Fort Donelson, Shiloh, New Orleans), which would in turn prevent several more battles that the CSA lost on OTL (Memphis, Stones River, Vicksburg...) from even taking place. That those ATL victories would prevent those subsequent OTL defeats is realistic, but six extra victories to begin with is a tall order. Not ASB, but very unlikely. Few people are that lucky. But the simple fact is: anything less would not be sufficient to make the Confederates look like total winners and the Union like total losers... and that precise impression is a prerequisite for European recognition of the CSA's independence.