I think it will be harder to design 1GHz x10 multiplier with good performance, x5 x2 feels much easier.
I would recommend to use 5ghz push-push oscillator + 5GHz frequency prescaler IC, and get much cleaner 10GHz without a "comb" of 1GHz,2GHZ,3GHZ,... harmonics
Also for x5 x2 case it may be easier to filter out comb at 10ghz after x5 stage, and then very simple to filter out 15GHz after x2 stage.
With x10 multiplier you will have ..7,8,9,10,11,12 GHz.. and maybe even worse if previous 0.1GHz x 10 = 1GHz have 0.7,0.8,0.9 GHz comb.
If you already have a good clean 1GHz, for x5 x2 approach i would use some FET, for example ne3210s01.
1) 1GHz -> ne3210s01 -> microstrip filter -> 5GHz -> ne3210s01 -> microstrip filter -> 10GHz (each fet have tunable negative bias on gate)
If there are some stability problems
2) 1GHz -> almost no gain ne3210s01 + negative bias (multiplier) -> filter -> ne3210s01 with gain ->
5GHz -> almost no gain ne3210s01 + negative bias (multiplier) -> filter -> 10GHz
optimize x5 and x2 stages by tuning negative bias
for 1 to 5GHz maybe use something else, because ne3210s01 may oscillate. For 5GHz to 10GHz ne3210s01 or similar will be good (thin substrate, tightly placed vias from source to ground, and 0.2mm wide quarterwave coupled lines with 0.2mm gap at 5 and 10 ghz).