Additional factor to be considered ( determines the subsequent character of the storm ):
* vertical wind shear through the depth of the convective layer.
(For more on this, see the article "CAPE, Shear, and the Thunderstorm")
The approach to forecasting these depends upon whether you are
(a): looking for 'individual' cellular storms, and associated phenomena, e.g. multi-cell, super-cell etc. OR ...
(b): vigorous convective activity embedded in layer cloud, frontal cloud etc.
1. Check for a trough (or a closed upper low) in the upper flow at around 700 or 500 hPa. If the trough is 'broadly' rounded at it's base (or you are dealing with a slow-moving, closed upper low), then convective activity can be expected across a reasonably wide area associated with this trough / upper cold pool; if the trough is moving; is 'sharper' at it's base, with a well-defined axis, then activity is roughly along and on the eastern (or 'forward') side of that axis. [ "eastern side" assumes that the trough is a classical west-to-east moving feature at mid-latitudes - other arrangements can and do occur, for example 'Easterly Waves' in tropical latitudes. ]
2. Once you've decided that the upper air charts are broadly conducive to such activity: use radiosonde ascents to assess items: 1, 3 and 4 above. If the answer is 'yes' to all three, with either expected surface temperatures, or due to forced lifting via dynamic ascent (700hPa vertical motion), orographic ascent (sufficient horizontal flow across hills/mountains) etc., (item 5) .. move onto 3. below)
3. You can assess moisture available via the radiosonde ascents, but they are getting few and far between now. Humidity levels can be inferred from 850hPa output from models.
4. If the answer to all five conditions is 'yes' then thunder is likely. To decide between individual cellular storms, and the variety of multi-cell, supercell etc., see "How does a single-cell shower differ from a multi-cell thunderstorm, or even a 'supercell'?".
(NB: if 3. and 4. above are not met, or only marginal, then a TCU/+SHRA is more likely than CB/TS etc.)
Modern-day forecasting centres often don't work out the steps above: they use NWP output to view forecast ascents, output cloud top temperatures, heights, etc. In operational forecast offices, we have computer output indicating the vigour of the showers, plus cloud top temperatures (for convective types), explicit Convective top/depth forecasts and forecast ascents: we use all these to define where/whether thunder will occur. Not all (if any) of these output are available on the Internet.
Another approach, favoured in North America, is to use instability indices: some of these can be seen on the plotted output from the University of Wyoming ... see the links from the article on thermodynamic diagrams.