Old Ball - welcome to the 'hood!
To be honest - anywhere in Chiswick will be alright and for 450 per week you can get a really nice flat - you'll be able to get a 2 bedder with that.
Will you have a car? You should be able to catch the bus to GSK anyway off the High Road.
In terms of areas the nicest areas are Bedford Park and Grove Park. Bedford Park is near to the tube (Chiswick Park and Turnham Green) and Grove Park is near the mainline train station and the river. Grove Park is on the opposite side of the A4 from the Chiswick High Road where all the shops, pubs and restaurants are. However, Grove Park is a bit more leafy and has more of a 'villagey' feel to it, and is close to the river and still has pubs and restaurants but is a little bit more away from the High Road (still only a 15 min walk to there though). Areas near to Gunnersbury tube can be patchy and also in and around Bollo Lane and Acton Lane - sometimes the outskirts of Acton get called Chiswick so watch out for that one. But it's all relative and to be honest there isn't really a dodgy part to Chiswick.
If you have any specific questions on an area or anything about moving here - please feel free to PM me and if you do move into the 'hood more than happy to meet up and show a fellow Wire the area.
I've had my flat here for over 10 years now and I wouldn't live anywhere else in London to be honest - it's a really nice place to live and doesn't feel like you're in a city but has all the benefits of it.
You can also see why my friends call me the Chiswick Tourist Board!!