How do I prove to my new unit that they should send me to WLC?

Improve your apft by 70 points

For starters, when on a military associated social media site, try using your actual name instead of what you have listed. In my twisted sense of thinking, your "Sneaka Dee" is meant in a twisted and perverted sense of humor. Which normally I wouldn't care too much about except when you are seeking advice from many a Soldier (same rank and way way higher) on here about leadership. And seeing the name Sneaka Dee just does not present professionalism/leadership to me.
Secondly, it is good to be hungry with your goals....but to show that you are ready for WLC (BLC now), you have to PROVE that you are ready and are the best candidate to send. And how do you prove that, well, you start by becoming that sponge again when you get to your duty assignment. Right now you have scholastic world on your brain.....you are about to be diving into operational world. Two entirely different beasts. Having SSD-1 complete is good since you can't go without it done. 230 PT score is good, but be ready to hear that the company standard for the unit you are heading to could very well be 260 or higher. I once knew a platoon sergeant in the Engineers where his standard before you could even smell the promotion board was an APFT score of 270. All of these things are good and all, but you still have to prove your worth to your new unit. And, as I said being hungry is good.......but if you show yourself to be too hungry then that could backfire on you. Be hungry, be driven, but don't be overbearing. Your new NCO's are going to be judging and evaluating you before you will even see BLC. Those NCOs are the first ones you will have prove yourself to.

You aren't ready. Just because you look good on paper means nothing. The quicker you understand that, the faster you'll go.

Show them by your actions.

I think it is a shame that so much emphasis is put on scoring a 300 on the APFT. One of my former BN XOs put it like this: who would you rather have on your team? a guy who is 6'2" 200 lbs that can ruck all day and carry a 249 in one hand but scores a 200 on the APFT or a guy who is 5'6" 130 pounds who maxes the APFT but can barely carry his own body weight? Each person is different, but simply because someone can score a 300 on the APFT, it doesn't guarantee they are going to be a good soldier.

Ask for responsibility.
Don't be an asshat.
Be the expert at your assigned tasks .
Volunteer to go to the Soldier of the Month board, actually study and do well at the board.
Lead the SOM and promotion board study group, if there is not one, ask to make one and participate.
Get better at PT, if it is perceived you are not that strong or look fluffy, fix it.
Help others with PT, volunteer to assist with the special population program. You get to do free extra PT and learn how to lead PT.
After a good bit (6-10 months) of showing you are ready to go to the next level, ask for it.

By your actions. Become a valuable asset and know your job inside and out.

Brown nose your leaders. Be really loud all the time and get a 300. Job competency and character don't matter in today's Army.

SGT(P) (Join to see)
First off, is your name really "Sneaka Dee"? If so that is fine, but remember that the Army is moving to be a more professional service. You are just getting ready to complete AIT, I admire your drive to become a leader and to progress in your career, but your leadership won't even look at sending you to BLC (what used to be WLC and PLDC) until you have proven yourself doing your job and have some time in the Army. Stay focused on your goals, become the "go to" guy. Start studying for the board now, volunteer somewhere once you get to your duty station and keep track of your hours, stay out of trouble and look for opportunities to do extra stuff for your unit. Work on getting your PT score at least in the 250 range, that way when you start competing in Soldier of the Quarter competitions, you won't lose points (preferable, in the 270 or above range to be competitive).

Don't stress, you sound like you have a good head on your shoulders. Keep working at your goals, don't let your leadership lose sight of YOUR goals (mention them in your monthly counselings) and you will be fine! Good luck with all that you do in the service.

Go to War first, then come back and see if you still give a shit. Not saying, just saying.

My recommendation? Don't rush it! You just learned the schoolhouse way of doing your job. Now go into the force and implement it. Learn from your peers, leaders, and civilian counterparts. Use your time to observe mission accomplishment, company level and below operations, and your leaders actions (and inactions). Volunteer for different duties and details to broaden your understanding ad learn what does and does not work. As stated many times, be at the right place, at the right time, and in the right uniform. Be present in all activities, open to constructive criticism, and engaged. Keep your bearing and bring a positive attitude. If you do that, your unit will be eager to send you to the board and NCOES. Just don't expect it to happen right away - this is truly about the journey and not the destination. Good luck!