The connection is varying or fluctuating. If it sometimes dies entirely, it is intermittent.
There's a few informal phrases that are often used.
Sometimes people use choppy by metaphorical reference to choppy weather at sea, that causes sharp waves. Turbulent by comparison to turbulence experienced in airplanes too.
Laggy just referring to the lag caused by the slow periods. This is not specific to a connection that's changeable, but also covers one that is always slow, though in practice it does tend to be one with slow periods; you'd have just given up trying to use it if it was always too slow.
In the context gaming, there are various joking forms of Lag such as Laaa…………aag, and references to the mysterious "Lag-monster" (who eats TCP/IP packets, hence causing the lag).
One sometimes also gets plossy as short for "packet loss" + an -y to make it an adjective, since packet loss is something that will cause poor network connections of the sort especially noticeable in gaming. This last is very much gamer-slang or hacker-slang and wouldn't be understood generally (whereas with the others, people could work out what you meant). A plossy connection is different to a slow one, as while the effect on a download is just to slow it (the lost packets get resent), in a live-action use the effect of a plossy fast connection is worse than that of a steady slow connection, resulting in the sort of freezing of your character that you describe.