Working up to a pull up?

One of my big fitness goals for myself over the next 5 months is to finally, for the first time in my life, be able to do an unassisted pullup. I never have eaked out even one without a jump start, or a machine. For that matter, I can only do approximately 5 "normal" pushups, and I'm pretty sure I don't go far enough to the ground! I am a 140lb 5'5 female, if this matters.

What are some good exercises to start building up the strength to do this? I am doing KettleWorx DVDs three times a week (which are essentially 25 minute workouts with kettlebells, split between resistance, cardio, and core), but I know this is definitely not enough. Since I am very new in my fitness journey, any input would be greatly appreciated!

Oh, and if this is important- I don't currently have access to a gym, but would be willing to spend anywhere under 100$ in equipment, if it would help. One thing that I plan to buy soon is a bar for weights (I believe those weigh 45lbs?) to start doing squats with- if there are any exercises that could help with getting up to a pullup, AWESOME.