Do you mean Asian as in curries and that sort of thing?
Where I live, Harehills, it's FULL of asian food places. You can buy loads of spices and halal meat and all sorts here. Harehills has a very large Asian community (not Chinese Asian, but Islamic Asian) so there are loads of shops to cater for their sort of cuisine.
It's about 15 minutes from Leeds city centre on the bus - just catch the number 49, 50, or 50A bus outside 'If' bar in the Light on the Headrow, and tell the driver you want to get off just past Kwik Save in Harehills.
Ask him to tell you where to get off.
Once you get off the bus, just have a look around the area. There's lots of places to buy spices. If you get stuck, just stop a passer by- people will tell you where to go I'm sure.
I'm sorry there's nowhere closer to Headingley though that I can think of for that kind of thing.