Moving Out, Saving help

So I'm planning on moving into a one bedroom apartment with just my daughter and myself. I'm not sure if this is the right group to post this in but I was hoping for some advice. I work in retail so automatically I get paid sucky. I'm hoping on moving by June 1st, and LO birthday is May 28. I don't want to have to cancel her party just to save for a apartment. I've never had a party for her since she's been born and since she arrived before my Babyshower I had a homecoming and no one showed up. 
Any tips you can give me to save?  I'm hoping to save about $1200-2,000 security deposit. What percent of my check do you recommend I put into savings? Do I put my whole check into my savings account?
Also if I offer to pay more than the asking security deposit is that good? Will it help anything?