Usually a fixed fee, often calculated on the purchase price and may be add ons depending on if it's a leasehold or something with more work involved.
Just make sure they give you a full illustration which includes Land Registry fees, any stamp duty and any disbursements or additional fees like TT's.
Definitely ring round, find someone who uses email as this will speed things up no end (the more old fashioned sorts who can't get a letter out without dictating it, waiting for a secretary to type it, checking it then waiting for it to arrive by post/DX just slows things down unecessarily.
It just makes for far quicker transactions, you can e.mail the seller's sols and copy the client and agent in in one go and forward replies, HIPS are usually online and documents like transfers and mortgage deeds can be e.mailed.
As if they have a policy about fees if the transaction doesn't proceed as some will not charge you if it doesn't complete, you would only pay for disbursements which are much lower now HIPS are involved.
I might be biased as it's what I do but I really would recommend using a lawyer as there are things you may not know or pick up on which could cause problems further down the line.
Many solicitors would also be wary of dealing with a private individual who is not represented and, if you are getting a mortgage, the lender would want legal advisors to ensure all that would be done properly. It's like they would charge you for it so you may as well get a legal advisor who does the whole thing.
As a guide my legal fees on a purchase for �100k would be �345.00 plus VAT and disbursements.