The problem with London is that it is so big and there is almost too much choice. With only 5 days, you may find that you won't have enough time to go to all of these but they all have something to offer. I will confine the suggestions to central London (more or less) but there are good markets everywhere so you may find others that you like near where you will be staying.
Ok Here goes:
For food:
Borough Market SE 1. This is a huge food market that is really expanding. Mostly people go to buy good quality and organic food, but if you are looking to buy food as gifts (such as ready packed shortbread etc,) I'm not sure how much you would find here, but the market has a great reputation and should be worth a visit for the experience.
Spitalfields market also has food + lots of other things and is an area undergoing regeneration (such as what happened to Covent Garden some years ago).
Berwick Street Market is in the heart of Soho and is used by people who work in the area for local shopping. Again, good for groceries, but no necessarily gift food. There are however loads of little shops very near the market which might have what you want - mind you a lot are Italian, so you are unlikely to be interested unless you are working on the coals to Newcastle principle.
Finally, apart from the Selfridges food hall which you've ruled out (but which is always worth a visit) there is a supermarket chain here called Waitrose and they have very good quality food of every description. In fact, I seem to recall that Waitrose won an award for its offering of cheeses. There is a new Waitrose store that has just opened in Docklands which has been set up specifically to cater to the wealthy city dealers. It is supposed to have things there that are not available elsewhere, but I have not yet been in to have a look. In any case, I am guessing that I would be unwilling to pay the prices for these elite foods, but again, it might be fun to have a look.
Antiques:
The most important one which all the dealers use is the Bermondsey market. It operates on Fridays only from about 5 am to 9 am. If you are interested in bargains, I gather you really have to get there at 5 am because the dealers clear out pretty early. This is the place where, in the past, Rembrandts have turned up for sale as unknown paintings. This was because there used to be a law, recently changed, that gave special status to certain markets (called 'Market Ouvert' I think) which meant that if you bought goods that turned out to be stolen, you were entitled to keep them.
Next for antiques, I would go to Camden Passage in Islington. This is only just outside the inner London circle and there are loads of other shops around the market selling non antique things that you might be interested it.
Finally, Portobello Market in Notting Hill is worth a visit, but the area has now been so hyped that you will probably have to look quite hard for a bargain.
Clothes:
Try Camden Market in Camden Town. This is a huge sprawling market that has masses of retro clothes, wholesale leathers, jeans galore etc etc. This also sells, some antiques, bric a brac and has lots of shops nearby. The only problem with Camden Market is that it has become a bit of a zoo, but nevertheless fun.
Petticoat Lane near Liverpool Street is a famous London market which is supposed to specialise in clothes, although I have to admit I haven't been there for years and years.
Flea Markets:
Has to be Brick Lane on a Sunday morning (you could combine a vist here with a visit to Petticoat Lane). I don't know if you'd find presents here or not, but this is a great east London institution and really interesting. Not too far away is Columbia Road market which specialises in plants and flowers which are sold off quite cheaply from about 1:30 because the market has to close down by 2 pm and the traders want to get rid of stock.
Here are some websites that give specifics regarding addresses, times etc.
Markets in LondonAnother markets listExplore London websiteI hope this helps, but let me know if you need any further help.
Beebee